discrete_mathematics_with_applications/chapter_3/exercises.md

Exercise Set 3.1

Page 142

1. A menagerie consists of seven brown dogs, two black dogs, six gray cats, ten black cats, five blue birds, six yellow birds, and one black bird. Determine which of the following statements are true and which are false.

a. There is an animal in the menagerie that is red.

False

b. Every animal in the menagerie is a bird or a mammal.

True

c. Every animal in the menagerie is brown or gray or black.

False

d. There is an animal in the menagerie that is neither a cat nor a dog.

True

e. No animal in the menagerie is blue.

False

f. There are in the menagerie a dog, a cat, and a bird that all have the same color.

True

2. Indicate which of the following statements are true and which are false. Justify your answers as best you can.

a. Every integer is a real number.

True, because the set of all integers, \mathbb{Z} is a subset of the set of all real numbers, \mathbb{R}. \mathbb{Z} \in \mathbb{R}.

b. 0 is a positive real number.

False, 0 is neither positive nor negative.

c. For every real number r, -r is a negative real number.

False, if r is negative, then -r is positive.

d. Every real number is an integer.

False, \dfrac{1}{2} is not an integer, but is a real number.

3. Let R(m, n) be the predicate "If m is a factor of n^2 then m is a factor of n," with domain for both m and n being \mathbb{Z} the set of integers.

a. Explain why R(m, n) is false if m = 25 and n = 10.

The statement "If 25 is a factor of 100" is a true hypothesis, but the conclusion "then 25 is a factor of 10" is false because 10 is not a product of 25 times any integer. Thus the hypothesis is true, but the conclusion is false, making this predicate a false statement.

b. Give values different from those in part (a) for which R(m, n) is false.

m = 9, n = 3

c. Explain why R(m, n) is true if m = 5 and n = 10.

Because 5 is a factor of 100, which is the hypothesis, and the conclusion is that 5 is a factor of 10, which is also true. Thus the hypothesis and conclusion are both true, so the statement as a whole is true.

d. Give values different from those in part c for which R(m, n) is true.

m = 4, n = 8

4. Let Q(x, y) be the predicate "If x < y then $x^2 < y^2$" with the domain for both x and y being \mathbb{R} the set of real numbers.

a. Explain why Q(x, y) is false if x = -2 and y = 1.

The hypothesis becomes "If $-2 < 1$", which is true, the conclusion becomes "then $4 < 1$", which is a false conclusion. Thus the hypothesis is true, but the conclusion is false, making Q(-2, 1) a false statement.

b. Give values different from those in part (a) for which Q(x, y) is false.

x = -5, y = 2

c. Explain why Q(x, y) is true if x = 3 and y = 8.

The hypothesis becomes "If $3 < 8$", which is true, and the conclusion becomes "then $9 < 64$", which is also true. This shows that the hypothesis and conclusion true, making Q(3, 8) a true statement.

d. Give values different from those in part c for which Q(x, y) is true.

x = 3, y = 4

5. Find the truth set of each predicate.

a. Predicate: \dfrac{6}{d} is an integer, domain: \mathbb{Z}

 \{-6, -3, -2, -1, 1, 2, 3, 6\} 

b. Predicate: \dfrac{6}{d} is an integer, domain: \mathbb{Z}^+

 \{1, 2, 3, 6\} 

c. Predicate: 1 \leq x^2 \leq 4, domain: \mathbb{R}

 \{x \in \mathbb{R} | -1 \leq x \leq -2 \text{ or } 1 \leq x \leq 2\} 

d. Predicate: 1 \leq x^2 \leq 4, domain: \mathbb{Z}

 \{-2, -1, 1, 2\} 

6. Let B(x) be "-10 < x < 10." Find the truth set of B(x) for each of the following domains.

a. \mathbb{Z}

 \{-9, -8, -7, -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9\} 

b. \mathbb{Z}^+

 \{1, 2, 3, 4, 5, 6, 7, 8, 9\} 

c. The set of all even integers

 \{-8, -6, -4, -2, 2, 4, 6, 8\} 

7. Let S be the set of all strings of length 3 consisting of a's, b's, and c's. List all the strings in S that satisfy the following conditions:

   1. Every string in S begins with b.

   baa, bab, bac, bba, bbb, bbc, bca, bcb, bcc

   2. No string in S has more than one c.

   aaa, aab, aac, aba, abb, abc, aca, acb, baa, bab, bac, bba, bbb, bbc, bca, bcb, caa, cab, cba, cbb

8. Let T be the set of all strings of length 3 consisting of 0's and 1's. List all the strings in T that satisfy the following conditions:

   1. For every string s in T, the second character of s is 1 or the first two characters of s are the same.

   000, 001, 010, 110, 111

   2. No string in T has all three characters the same.

   001, 010, 100, 101, 110

Find counterexamples to show that the statements in 9-12 are false.

9. \forall x \in \mathbb{R}, x \geq \dfrac{1}{x}.

x = \dfrac{1}{2}, so \dfrac{1}{2} \geq \dfrac{1}{\dfrac{1}{2}} is false as:

 \frac{1}{2} < 2 

10. \forall a \in \mathbb{Z}, \dfrac{(a - 1)}{a} is not an integer.

 a = -1 

 \frac{((-1) - 1)}{-1} = \frac{-2}{-1} = 2 

Since 2 \in \mathbb{Z}, this statement is false.

11. \forall positive integers m and n, m \cdot n \geq m + n.

m = 1, n = 2

 1 \cdot 2 = 2 \geq 3 = 1 + 2 

But 2 is not greater than or equal to 3, so this statement is false.

12. \forall real numbers x and y, \sqrt{x + y} = \sqrt{x} + \sqrt{y}.

x = 4, y = 9

 \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275 

 \sqrt{4} + \sqrt{9} = 2 + 3 = 5 

Since \sqrt{13} \neq 5, this statement is false.

13. Consider the following statement:

\forall basketball player x, x is tall.

Which of the following are equivalent ways of expressing the statement?

a. Every basketball player is tall.

Yes.

b. Among all the basketball players, some are tall.

No.

c. Some of all the tall people are basketball players.

No.

d. Anyone who is tall is a basketball player.

No.

e. All people who are basketball players are tall.

Yes.

f. Anyone who is a basketball player is a tall person.

Yes.

14. Consider the following statement:

\exists x \in \mathbb{R} such that x^2 = 2.

Which of the following are equivalent ways of expressing this statement

a. The square of each real number is 2.

No.

b. Some real numbers have square 2.

Yes.

c. The number x has square 2, for some real number x.

Yes.

d. If x is a real number, then x^2 = 2.

No.

e. Some real number has square 2.

Yes.

f. There is at least one real number whose square is 2.

Yes.

15. Rewrite the following statements informally in at least two different ways without using variables or quantifiers.

a. \forall rectangle x, x is a quadrilateral.

If a shape is a rectangle, then the shape is a quadrilateral.

For any given rectangle, that rectangle is a quadrilateral.

b. \exists a set A such that A has 16 subsets.

There must be a set that has 16 subsets.

At least one set has 16 subsets.

16. Rewrite each of the following statements in the form "\forall ______ x, ______."

a. All dinosaurs are extinct.

\forall dinosaurs, x, x is extinct.

b. Every real number is positive, negative, or zero.

\forall real numbers x, x is positive, negative, or zero.

c. No irrational numbers are integers.

\forall irrational numbers x, x is not an integer.

d. No logicians are lazy.

\forall logicians x, x is not lazy.

e. The number 2,147,581,953 is not equal to the square of any integer.

\forall integer x, x^2 does not equal 2,147,581,953.

f. The number -1 is not equal to the square of any real number.

\forall real numbers x, x^2 does not equal -1.

17. Rewrite each of the following in the form "\exists ______ x such that ______."

a. Some exercises have answers.

\exists some exercise, x, such that x has an answer.

b. Some real numbers are rational.

\exists some real number x, such that x is rational.

18. Let D be the set of all students at your school, and let M(s) be "s is a math major," let C(s) be "s is a computer science student," and let E(s) be "s is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates M(s), C(s), and E(s).

a. There is an engineering student who is a math major.

\exists s such that E(s) and M(s).

b. Every computer science student is an engineering student.

\forall s \in D, C(s) \to E(s)

c. No computer science students are engineering students.

\forall s \in D, C(s) \to \neg E(s)

d. Some computer science students are also math majors.

\exists s such that C(s) \wedge M(s)

e. Some computer science students are engineering students and some are not.

\exists s such that C(s) \wedge E(s) or \exists such that C(s) \wedge \neg E(s)

19. Consider the following statement:

\forall integer n, if n^2 is even then n is even.

Which of the following are equivalent ways of expressing this statement?

a. All integers have even squares and are even.

No

b. Given any integer whose square is even, that integer is itself even.

Yes

c. For all integers, there are some whose square is even.

No

d. Any integer with an even square is even.

Yes

e. If the square of an integer is even, then that integer is even.

Yes

f. All even integers have even squares.

No

20. Rewrite the following statement informally in at least two different ways without using variables of the symbol \forall or the words "for all."

\forall real numbers x, if x is positive then the square root of x is positive.

If a number is a positive real number, then the square root of that number is positive.

Any positive real number's square root is positive.

21. Rewrite the following statements so that the quantifier trails the rest of the sentence.

a. For any graph G, the total degree of G is even.

The total degree of G is even, for any graph G.

b. For any isosceles triangle T, the base angles of T are equal.

The base angles of T are equal, for any isosceles triangle T.

c. There exists a prime number p such that p is even.

p is even for some prime number p.

d. There exists a continuous function f such that f is not differentiable.

f is not differentiable for some continuous function f.

22. Rewrite each of the following statements in the form "\forall ______ x, if ______ then ______."

a. All Java programs have at least 5 lines.

\forall programs x, if x is a Java program, then it has at least 5 lines.

b. Any valid argument with true premises has a true conclusion.

\forall arguments x, if x is a valid argument with a true premise, then it has a true conclusion.

23. Rewrite each of the following statements in the two forms "\forall x, if ______ then ______" and "\forall x, ______" (without an if-then).

a. All equilateral triangles are isosceles.

\forall x, if x is equilateral, then x is isosceles.

\forall equilateral triangles x, x is isosceles.

b. Every computer science student needs to take data structures.

\forall x if x is a computer science student, then x needs to take data structures.

\forall computer science students x, x needs to take data structures.

24. Rewrite the following statements in the two forms "\exists ______ x such that ______" and "\exists x such that ______ and ______."

a. Some hatters are mad.

\exists a hatter x such that x is mad.

\exists x such that x is a hatter and x is mad.

b. Some questions are easy.

\exists a question x such that x is easy.

\exists x such that x is a question and x is easy.

25. The statement "The square of any rational number is rational" can be rewritten formally as "For all rational numbers x, x^2 is rational" or as "For all x, if x is rational then x^2 is rational." Rewrite each of the following statements in the two forms "\forall ______ x, ______" and "\forall x, if ______, then ______" or in the two forms "\forall ______ x and y, ______" and "\forall x and y, if ______, then ______."

a. The reciprocal of any nonzero function is a fraction.

\forall nonzero function x, the reciprocal of x is a fraction.

\forall x, if x is a nonzero fraction, then the reciprocal of x is a fraction.

b. The derivative of any polynomial function is a polynomial function.

\forall derivatives of any polynomial function x, x is a polynomial function.

\forall x, if x is a derivative of any polynomial function, then x is a polynomial function.

c. The sum of the angles of any triangle is 180\degree.

\forall triangles x, the sum of the angles of x is 180\degree.

\forall x if x is a triangle, then the sum of the angles of x is 180\degree.

d. The negative of any irrational number is irrational.

\forall negative of any irrational number, x, x is irrational.

\forall x if x is a negative of any irrational number, then x is irrational.

e. The sum of any two even integers is even.

\forall even integers x, and y, the sum of x and y is even.

\forall x, y if x and y are even integers, then the sum of x and y is even.

f. The product of any two fractions is a fraction.

\forall fractions x and y, the product of x and y is a fraction.

\forall x, y if x and y are fractions, then the product of x and y is a fraction.

26. Consider the statement "All integers are rational numbers but some rational numbers are not integers."

a. Write this statement in the form "\forall x, if ______ then ______, but \exists ______ x, such that ______."

\forall x, if x is an integer, then x is a rational number, but \exists a rational number x, such that x is not an integer.

b. Let \text{Ratl}(x) be "x is a rational number" and \text{Int}(x) be "x is an integer." Write the given statement formally using only the symbols \text{Ratl}(x), \text{Int}(x), \forall, \exists, \wedge, \vee, \neg, and \to.

 \forall x (\text{Int}(x) \to \text{Ratl}(x)) \wedge \exists x (\text{Ratl(x)} \wedge \neg \text{Int}(x))

27. Refer to the picture of Tarski's world given in Example 3.1.1.3. Let \text{Above}(x, y) mean that x is above y (but possibly in a different column). Determine the truth or falsity of each of the following statements. Give reasons for your answers.

a. \forall u, \text{Circle}(u) \to \text{Gray(u)}.

This is false, b is a circle and is black.

b. \forall u, \text{Gray}(u) \to \text{Circle}(u).

This is true, all gray shapes are circles.

c. \exists y such that \text{Square}(y) \wedge \text{Above}(y, d).

This is false, there is no shape that is a square and is above shape d.

d. \exists z such that \text{Triangle}(z) \wedge \text{Above}(f, z).

This is true, shape g is a triangle where shape f is above shape g.

In 28-30, rewrite each statement without using quantifiers or variables. Indicate which are true and which are false, and justify your answers as best as you can.

28. Let the domain of x be the set D of objects discussed in mathematics courses, and let \text{Real}(x) be "x is a real number," \text{Pos}(x) be "x is a positive real number," \text{Neg}(x) be "x is a negative real number," and \text{Int}(x) be "x is an integer."

a. \text{Pos}(0)

"0 is a positive real number."

This is a false statement, as 0 is neither positive nor negative.

b. \forall x, \text{Real}(x) \wedge \text{Neg}(x) \to \text{Pos}(-x)

"For any number, if that number is both real and negative, then the negative of that number is positive.""

This is true, if you take the negative of a negative of any real number, then it is positive.

c. \forall x, \text{Int}(x) \to \text{Real}(x)

"For any number, if that number is an integer, then that number is a real number."

This is true, the set of all integers is a subset of all real numbers.

d. \exists x such that \text{Real}(x) \wedge \neg \text{Int}(x)

"There is at least one number that is both a real number and not an integer."

This is true, an example would be \dfrac{1}{2}, which is a real number but not an integer.

29. Let the domain of x be the set of geometric figures in the plane, and let \text{Square}(x) be "x is a square" and \text{Rect}(x) be "x is a rectangle."

a. \exists x such that \text{Rect}(x) \wedge \text{Square}(x)

"There exists a shape that is both a rectangle and a square."

This is true since any shape that is a square is also a rectangle.

b. \exists x such that \text{Rect}(x) \wedge \neg \text{Square}(x)

"There exists a shape that is both a rectangle and not a square."

This is true, as any shape that is a rectangle that has unequal length and width is not a square.

c. \forall x, \text{Square}(x) \to \text{Rect}(x)

"For any shape, if that shape is a square, then that shape is a rectangle."

This is true, for all shapes, any shape that is a square, that shape is then a rectangle.

30. Let the domain of x be \mathbb{Z}, the set of integers, and let \text{Odd}(x) be "x is odd," \text{Prime}(x) be "x is prime," and \text{Square}(x) be "x is a perfect square." (An integer n is said to be a perfect square if, and only if, it equals the square of some integer. For example, 25 is a perfect square because 25 = 5^2.)

a. \exists x such that \text{Prime}(x) \wedge \neg \text{Odd}(x)

"There exists some number that is both prime and not odd."

This is true, for example 2 is a prime number (cannot be divided except by 1 and itself), but 2 is also not odd.

b. \forall x, \text{Prime}(x) \to \neg \text{Square}(x)

"For any number, if that number is prime, then that number is not a perfect square."

This is true, since a prime number is only divisible by 1 and itself, it cannot equal the square of some integer, since that square would also be the product of two smaller positive integers.

c. \exists x such that \text{Odd}(x) \wedge \text{Square}(x)

"There exists some number that is both odd and is a perfect square."

This is true, take 9 as an example, 9 is an odd number, but is also a perfect square as 9 = 3^2.

31. In any mathematics or computer science text other than this book, find an example of a statement that is universal but is implicitly quantified. Copy the statement as it appears and rewrite it making the quantification explicit. Give a complete citation for your example, including title, author, publisher, year, and page number.

Omitted.

32. Let \mathbb{R} be the domain of the predicate variable x. Which of the following are true and which are false? Give counter examples for the statements that are false.

a. x > 2 \Rightarrow x > 1

This is true, for any real number that is greater than 2, that same real number is greater than 1.

b. x > 2 \Rightarrow x^2 > 4

This is true, for any real number that is greater than 2, that same real number squared is greater than 4.

c. x^2 > 4 \Rightarrow x > 2

This is false, as x = -3 would mean (-3)^2 > 4, which is true as that is 9 > 4, but then (-3) > 2 is false. Since the hypothesis is true, but the conclusion is false for at least one example, this predicate is therefore false.

d. x^2 > 4 \Leftrightarrow |x| > 2

This is true. For all numbers x, if x^2 > 4, then |x| > 2 is true.

Additionally, for all numbers x, if |x| > 2, then x^2 > 4 is true.

Since both directions of this universal "if and only if" statement are true, this is a true statement.

33. Let \mathbb{R} be the domain of the predicate variables a, b, c, and d. Which of the following are true and which are false? Give counterexamples for the statements that are false.

a. a > 0 \text{ and } b > 0 \Rightarrow ab > 0

This is true. If both a and b are positive, then their product is also positive.

b. a < 0 \text{ and } b < 0 \Rightarrow ab < 0

This is false, If both a and b are negative, then their product is positive, not negative. Take -1 and -2 for example, whose product is 2.

c. ab = 0 \Rightarrow a = 0 \text{ or } b = 0

This is true, for all real numbers a and b, if their product, ab is equal to 0, then either a or b must be 0.

d. a < b \text{ and } c < d \Rightarrow ac < bd

This is false. Say a = -1, b = 2, c = -8 and d = 3. This would make -1 < 2 and -8 < 3, which is true, but then (-1)(-8) < (2)(3) would be 8 < 6, which is false.

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Exercise Set 3.2

Page 152

1. Which of the following is a negation for "All discrete mathematics students are athletic"? More than one answer may be correct.

a. There is a discrete mathematics student who is nonathletic.

Yes.

b. All discrete mathematics students are nonathletic.

No.

c. There is an athletic person who is not a discrete mathematics student.

No.

d. No discrete mathematics students are athletic.

No.

e. Some discrete mathematics students are nonathletic.

Yes.

f. No athletic people are discrete mathematics students.

No.

2. Which of the following is a negation for "All dogs are loyal"? More than one answer may be correct.

a. All dogs are disloyal.

No.

b. No dogs are loyal.

No.

c. Some dogs are disloyal.

Yes.

d. Some dogs are loyal.

No.

e. There is a disloyal animal that is not a dog.

No.

f. There is a dog that is disloyal.

Yes.

g. No animals that are not dogs are loyal.

No.

h. Some animals that are not dogs are loyal.

No.

3. Write the formal negation for each of the following statements.

a. \forall string s, s has at least one character.

\exists a string s, such that s does not have any characters.

b. \forall computer c, c has a CPU.

\exists a computer c, such that c does not have a CPU.

c. \exists a movie m such that m is over 6 hours long.

\forall movies m, m is no longer than 6 hours long.

d. \exists a band b such that b has won at least 10 Grammy awards.

\forall bands b, b has won at most 9 Grammy awards.

4. Write an informal negation for each of the following statements. Be careful to avoid negations that are ambiguous.

a. All dogs are friendly.

There is at least one dog that is not friendly.

b. All graphs are connected.

There is at least one graph that is not connected.

c. Some suspicions were substantiated.

All suspicions were unsubstantiated.

d. Some estimates are accurate.

No estimates are accurate.

5. Write a negation for each of the following statements.

a. Every valid argument has a true conclusion.

There exists at least one valid argument that has a false conclusion.

b. All real numbers are positive, negative, or zero.

There exists at least one real number that is neither positive, negative, nor zero.

Write a negation for each statement in 6 and 7.

a. Sets A and B do not have any points in common.

Sets A and B have at least one point in common.

b. Towns P and Q are not connected by any road on the map.

There exists at least one road on the map that connects towns P and Q.

a. This vertex is not connected to any other vertex in the graph.

All vertexes are connected to at least one other vertex in the graph.

b. This number is not related to any even number.

All numbers are related to at least one even number.

8. Consider the statement "There are no simple solutions to life's problems." Write an informal negation for the statement, and then write the statement formally using quantifiers and variables.

"There is at least one of life's problems for which there is a simple solution."

\exists some life problem x, for which there is a simple solution.

Write a negation for each statement in 9 and 10.

9. \forall real number x, if x > 3 then x^2 > 9.

\exists a real number x, such that x > 3 and x^2 \leq 9.

10. \forall computer program P, if P compiles without error messages, then P is correct.

\exists a computer program P, such that P compiles without error messages and P is incorrect.

In each of 11-14 determine whether the proposed negation is correct. If it is not, write a correct negation.

Statement: The sum of any two irrational numbers is irrational.

Proposed negation: The sum of any two irrational numbers is rational.

No.

Corrected negation: There are at least two irrational numbers whose sum is rational.

Statement: The product of any irrational number and any rational number is irrational.

Proposed negation: The product of any irrational number and any rational number is rational.

No.

Corrected negation: There is at least one irrational number and at least one rational number whose product is rational.

Statement: For every integer n, if n^2 is even then n is even.

Proposed negation: For every integer n, if n^2 is even then n is not even.

No.

Corrected negation: There exists an integer n such that n^2 is even and n is not even.

Statement: For all real numbers x_1 and x_2, if x_1^2 = x_2^2 then x_1 = x_2.

Proposed negation: For all real numbers x_1 and x_2, if x_1^2 = x_2^2 then x_1 \neq x_2.

No.

Corrected negation: There exists two real numbers x_1 and x_2 such that x_1^2 = x_2^2 and x_1 \neq x_2.

15. Let D = \{-48, -14, -8, 0, 1, 3, 16, 23, 26, 32, 36\}. Determine which of the following statements are true and which are false. Provide counterexamples for the statements that are false.

a. \forall x \in D, \text{ if } x \text{ is odd then } x > 0.

True. All odd numbers in D are positive.

b. \forall x \in D, \text{ if } x \text{ is less than } 0 \text{ then } x \text{ is even}.

True. All negative numbers in D are even.

c. \forall x \in D, \text{ if } x \text{ is even then } x \leq 0.

False. There are some numbers in D that are even and greater than 0.

 \exists x \in D, x \text{ is even } \wedge x > 0 

The numbers 16, 26, 32, and 36 are counterexamples that show that this statement is false.

d. \forall x \in D, \text{ if the ones digit of } x \text{ is } 2, \text{ then the tens digit is } 3 \text{ or } 4.

This is true. There is only one number that satisfies the hypothesis "the ones digit of x is 2", which is 32. The conclusion is "the tens digit is 3 or 4", and that is true of the number 32, so this is a true statement.

e. \forall x \in D, \text{ if the ones digit of } x \text{ is } 6, \text{ then the tens digit is } 1 \text{ or } 2.

This is false. Consider the negation statement:

 \exists x \in D, \text{ such that the one's digit of } x \text{ is } 6 \wedge \text{ the tens digit of } x \text{ is not } 1 \text{ or } 2  

The number 36 satisfies the negation statement, and so therefore the given statement is false.

In 16-23, write a negation for each statement.

16. \forall real number x, if x^2 \geq 1 then x > 0.

\exists a real number x, such that x^2 \geq 1 and x \leq 0.

17. \forall integer d, if \dfrac{6}{d} is an integer then d = 3.

\exists an integer d such that \dfrac{6}{d} is an integer and d \neq 3.

18. \forall x \in \mathbb{R}, if x(x + 1) > 0 then x > 0 or x < -1.

\exists x \in \mathbb{R} such that x(x + 1) > 0 and both x \leq 0 and x \geq -1.

19. \forall x \in \mathbb{Z}, if n is prime then n is odd or n = 2.

\exists x \in \mathbb{Z} such that n is prime and both n is even and n \neq 2.

20. \forall integers a, b, and c, if a - b is even and b - c is even, then a - c is even.

\exists integers a, b, and c such that a - b is even and b - c is even and a - c is not even.

21. \forall integer n, if n is divisible by 6, then n is divisible by 2 and n is divisible by 3.

\exists an integer n such that n is divisible by 6 and either n is not divisible by 2 or n is not divisible by 3.

22. If the square of an integer is odd, then the integer is odd.

\exists an integer with the property that the square of the integer is odd but the integer itself is not odd.

23. If a function is differentiable then it is continuous.

\exists a function that is differentiable but is not continuous.

24. Rewrite the statements in each pair in if-then form and indicate the logical relationship between them.

a.

All the children in Tom's family are female.

If the person is a child in Tom's family, then the person is female.

All the females in Tom's family are children.

If the person is a female in Tom's family, then the person is a child.

The second statement is the converse of the first.

b.

All the integers that are greater than 5 and end in 1, 3, 7, or 9 are prime.

If the integer is greater than 5 and ends in 1, 3, 7, or 9, then the integer is prime.

All the integers that are greater than 5 and are prime end in 1, 3, 7, or 9.

If the integer is greater than 5 and is prime, then the integer ends in 1, 3, 7, or 9.

The second statement is the converse of the first.

25. Each of the following statements is true. In each case write the converse statement, and give a counterexample showing that the converse is false.

a. If n is any prime number that is greater than 2, then n + 1 is even.

If n + 1 is even, then n is any prime number that is greater than 2.

Counterexample: Let n = 15, n + 1 = 16, which is not a prime number.

b. If m is an odd integer, then 2m is even.

If 2m is even, then m is an odd integer.

Counterexample: Let m = 2, 2(2) = 4, and 4 is not an odd integer.

c. If two circles intersect in exactly two points, then they do not have a common center.

If two circles do not have a common center, then the two circles intersect in exactly two points.

Counterexample: Consider two circles that simply do not touch or overlap at all, both circles do not have a common center and they do not intersect at exactly two points.

In 26-33, for each statement in the referenced exercise write the contrapositive, converse, and inverse. Indicate as best as you can which of these statements are true and which are false. Give a counterexample for each that is false.

26. Exercise 16

Original Statement:

\forall real number x, if x^2 \geq 1 then x > 0.

This is false. Consider x = -2, then the hypothesis 4 \geq 1 is true, but the conclusion -2 > 0 is false. Therefore this statement is false.

Contrapositive:

\forall real number x, if x \leq 0 then x^2 < 1.

This is false. Consider x = -2, then the hypothesis -2 \leq 0 is true, but the conclusion 4 < 1 is false. Therefore this statement is false.

Converse:

\forall real number x, if x > 0 then x^2 \geq 1.

This is false. Consider x = \dfrac{1}{2}, then the hypothesis \dfrac{1}{2} > 0 is true, but the conclusion \dfrac{1}{4} \geq 1 is false. Therefore this statement is false.

Inverse:

\forall real number x, if x^2 < 1 then x \leq 0.

This is false. Consider x = \dfrac{1}{2}, then the hypothesis \dfrac{1}{4} < 1 is true, but then the conclusion \dfrac{1}{2} \leq 0 is false. Therefore this statement is false.

27. Exercise 17

Original Statement:

\forall integer d, if \dfrac{6}{d} is an integer then d = 3.

This is false, consider d = 1 (6 would also work). The hypothesis \dfrac{6}{1} = 6 is an integer is true, but then the conclusion 1 = 3 is false. Therefore this statement is false.

Contrapositive:

\forall integer d, if d \neq 3 then \dfrac{6}{d} is not an integer.

This is false. Consider d = 6. The hypothesis d \neq 6 is true, but the conclusion \dfrac{6}{6} = 1 is not an integer is not true. Therefore this statement is false.

Converse:

\forall integer d, if d = 3 then \dfrac{6}{d} is an integer.

This is true, if d = 3, then \dfrac{6}{3} = 2 is an integer is true. Therefore this is a true statement.

Inverse:

\forall integer d, if \dfrac{6}{d} is not an integer then d \neq 3.

This is true, if \dfrac{6}{d} is not an integer, than d cannot be 3 as then \dfrac{6}{3} = 2 which is an integer. Therefore d \neq 3 is a true conclusion and this is a true statement.

28. Exercise 18

Original Statement:

\forall x \in \mathbb{R}, if x(x + 1) > 0 then x > 0 or x < -1.

This is true, the hypothesis can be true, and its conclusion cannot be false.

Contrapositive:

\forall x \in \mathbb{R}, if x \leq 0 and x \geq -1 then x(x + 1) \leq 0.

This is true, x must lie in between 0 and 1 or be 0 or 1, and that will always cause x(x + 1) \leq 0 to be either negative or 0. This is a true statement.

Converse:

\forall x \in \mathbb{R}, if x > 0 or x < -1 then x(x + 1) > 0.

This is true, this essentially means that x(x + 1) will always be positive, as x is either always positive, or x is always negative, but the multiplication in the conclusion causes x(x + 1) > 0 to always be true. This statement is true.

Inverse:

\forall x \in \mathbb{R}, if x(x + 1) \leq 0 then x \leq 0 and x \geq -1.

Yes this is true, if x(x + 1) is either negative or 0, then x must be between 0 and -1 inclusive. This statement is true.

29. Exercise 19

Original Statement:

\forall x \in \mathbb{Z}, if n is prime then n is odd or n = 2.

This is a true statement, all prime numbers except for 2 are odd.

Contrapositive:

\forall x \in \mathbb{Z}, if n is not odd and n \neq 2 then n is not prime.

This is a true statement, if n is even and not 2, then n is not a prime number.

Converse:

\forall x \in \mathbb{Z}, if n is odd or n = 2, then n is prime.

This is false, consider n = 9, the hypothesis that n is odd or n = 2 is true, but the conclusion n is prime is false as 9 is divisible by 3. This is a false statement.

Inverse:

\forall x \in \mathbb{Z}, if n is not prime then both n is not odd and n \neq 2.

This is false, consider n = 9, the hypothesis n is not prime is true, but the hypothesis n is not odd and n \neq 2 is false as 9 is an odd number. This statement is false.

30. Exercise 20

Original Statement:

\forall integers a, b, and c, if a - b is even and b - c is even, then a - c is even.

This statement is true, as an even difference between any two numbers means that the two numbers themselves are even, so therefore a, b, and c are even, and a - c is even. This statement is true.

Contrapositive:

\forall integers a, b, and c, if a - c is not even, then either a - b is not even or b - c is not even.

This statement is true. An odd difference means that at least one of the two integers is odd, but necessarily both, so there for if a - c is odd, then either a or c is odd (but not both). This means that the conclusion is true because it allows for either a - b or b - c to be odd, which must be true if either a or c must be odd. This statement is true.

Converse:

\forall integers a, b, and c, if a - c is even, then a - b is even and b - c is even.

This is false, consider a = 13, b = 10, and c = 11. This means the hypothesis a - c = 13 - 11 = 2 is even is true, but then the conclusion a - b = 13 - 10 = 3 is even and 10 - 11 = -1 is even is false. This statement is false.

Inverse:

\forall integers a, b, and c, if a - b is not even or b - c is not even, then a - c is not even.

This is false, consider a = 13, b = 10, and c = 11. This means the hypothesis a - b = 13 - 10 = 3 is not even or b - c = 10 - 11 = -1 is not even is true, but then the conclusion 13 - 11 = 2 is not even is false. This statement is false.

31. Exercise 21

Original Statement:

\forall integer n, if n is divisible by 6, then n is divisible by 2 and n is divisible by 3.

This is true, any integer divisible by a non-prime number is divisible by those integer's prime factors.

Contrapositive:

\forall integer n, if n is not divisible by 2 or n is not divisible by 3, then n is not divisible by 6.

This is true. If n is not divisible by the prime numbers 2 or 3, then it is not divisible by 6.

Converse:

\forall integer n, if n is divisible by 2 and n is divisible by 3, then n is divisible by 6.

This is true, for the same reasons given by the original statement.

Inverse:

\forall integer n, if n is not divisible by 6, then n is not divisible by 2 or n is not divisible by 3.

This is true, for the same reasons given in the contrapositive.

32. Exercise 22

Original Statement:

If the square of an integer is odd, then the integer is odd.

This is a true statement, if the square of an integer returns an odd integer then that integer is odd.

Contrapositive:

If an integer is not odd, then the square of the integer is not odd.

This is a true statement, if an integer is not odd, it's square is not odd (even).

Converse:

If an integer is odd, then the square of the integer is odd.

This is a true statement, squaring any odd integer returns an odd integer.

Inverse:

If the square of an integer is not odd, then the integer is not odd.

This is a true statement, if squaring an integer returns a not odd integer, then that integer is not odd.

33. Exercise 23

Original Statement:

If a function is differentiable then it is continuous.

This is true, as a function that is differentiable means that the function has a limit that does not approach $\infty$/-\infty, and a continuous function is any function without any breaks in it's graph along the standard Cartesian plane.

Contrapositive:

If a function is not continuous then it is not differentiable.

This is true. If a function has any breaks, it's slope cannot be calculated, and therefore the function is not differentiable.

Converse:

If a function is continuous, then it is differentiable.

This is false. A function can be continuous, but it's limit can approach $-\infty$/\infty, so therefore it could be not differentiable. This statement is false.

Inverse:

If a function is not differentiable then it is not continuous.

This is not true. Just because a function's limit can approach $\infty$/-\infty, meaning it is not differentiable, does not necessarily mean that it is not continuous, as its limit could still approach some fixed number. This statement is false.

34. Write the contrapositive for each of the following statements.

a. If n is prime, then n is not divisible by any prime number from 2 through \sqrt{n}. (Assume that n is a fixed integer.)

If n is divisible by any prime number from 2 through \sqrt{n} inclusive, then n is prime.

b. If A and B do not have any elements in common, then they are disjoint. (Assume that A and B are fixed sets.)

If A and B are not disjoint, then A and B have some elements in common.

35. Give an example to show that a universal conditional statement is not logically equivalent to its inverse.

Consider the statement "If a student studies math, then they wear glasses." This is a universal conditional statement. Consider the inverse statement. "If a student does not study math, then they do not wear glasses." These two statements are not logically equivalent. The original statement claims that all students who study math wear glasses, where the second statement claims that all students who don't study math do not wear glasses.

Consider two students, Alice and Bob. Alice studies math and wears glasses, while Bob does not study math, but wears glasses. For the original statement, these two cases would be true, Alice does study math and wears glasses, whereas Bob does not study math, so the conclusion that he doesn't wear glasses still is a true conclusion. Juxtapose this with the inverse statement, where Bob not studying math is true (a true hypothesis), but he does wear glasses, which negates the inverse statement's conclusion (it is false). This is a false statement, and therefore the original statement and the inverse statement are not logically equivalent.

36. If P(x) is a predicate and the domain of x is the set of all real numbers, let R be "$\forall x \in \mathbb{Z}, P(x)$" let S be "$\forall x \in \mathbb{Q}, P(x)$", and let T be "\forall x \in \mathbb{R}, P(x)."

a. Find a definition for P(x) (but do not use "$x \in \mathbb{Z}$") so that R is true and both S and T are false.

Consider P(x) = 3x \neq 1. This is true of R, as there is no integer x where 3x = 1, but is not false for S and T where x = \dfrac{1}{3}.

b. Find a definition for P(x) (but do not use "$x \in \mathbb{Q}$") so that both R and S are true and T is false.

This isn't possible, it might be possible for P(n, x), but with just a single variable P(x), this is not possible.

37. Consider the following sequence of digits: 0204. A person claims that all the 1's in the sequence are to the left of all the 0's in the sequence. Is this true? Justify your answer. (Hint: Write the claim formally and write a formal negation of it. Is the negation true or false?)

Formal claim:

In the sequence of digits 0204, all digits that are 1's are to the left of all the 0's.

Negation:

In the sequence of digits 0204, there exists at least one 1 that is not to the left of all 0's.

The negation claim is false because there is no 1 in the sequence, so the original claim is vacuously true.

38. True or false? All occurrences of the letter u in Discrete Mathematics are lowercase. Justify your answer.

Formal claim:

\forall letters x in the string Discrete Mathematics, if x is the letter u, then x is lowercase.

Negation:

\exists some letter x in the string Discrete Mathematics such that x is the letter u and x is not lowercase.

The negation statement is false, as there is no letter u in Discrete Mathematics, and so the original statement is vacuously true.

Rewrite each statement of 39-44 in if-then form.

39. Earning a grade of C- in this course is a sufficient condition for it to count toward graduation.

If a person earns a grade of C- in this course, then the course counts towards graduation.

40. Being divisible by 8 is a sufficient condition for being divisible by 4.

If a number is divisible by 8, then it is divisible by 4.

41. Being on time each day is a necessary condition for keeping this job.

If a person is not on time each day, then this person will not keep this job.

42. Passing a comprehensive exam is a necessary condition for obtaining a master's degree.

If a person does not pass a comprehensive exam, then they will not obtain a master's degree.

43. A number is prime only if it is greater than 1.

If a number is prime, then it is greater than 1.

44. A polygon is square only if it has four sides.

If a polygon is square, then it has four sides.

Use the fact that the negation of a \forall statement is a \exists statement and that the negation of an if-then statement is an and statement to rewrite each of the statements 45-48 without using the word necessary or sufficient.

45. Being divisible by 8 is not a necessary condition for being divisible by 4.

Original without not is:

Being divisible by 8 is a necessary condition for being divisible by 4.

As if-then:

If a number is not divisible by 8, then it is not divisible by 4.

Negation:

There is a number that is not divisible by 8 and is divisible by 4.

46. Having a large income is not a necessary condition for a person to be happy.

Original without not:

Having a large income is a necessary condition for a person to be happy.

If-then (necessary condition):

If a person does not have a large income, then they are not happy.

Negation:

There is a person that does not have a large income and is happy.

47. Having a large income is not a sufficient condition for a person to be happy.

Original without not:

Having a large income is a sufficient condition for a person to be happy.

If-not(sufficient):

If a person has a large income, then they are happy.

Negation:

There is a person with a large income that is not happy.

48. Being a polynomial is not a sufficient condition for a function to have a real root.

Original without not:

Being a polynomial is a sufficient condition for a function to have a real root.

If-then(sufficient):

If a function is a polynomial, then it has a real root.

Negation:

There is a function that is a polynomial and does not have a real root.

49. The computer scientists Richard Conway and David Gries once wrote:

The absence of error messages during translation of a computer program is only a necessary and not a sufficient condition for reasonable [program] correctness.

Rewrite this statement without using the words necessary or sufficient.

Divide it up into two statements:

The absence of error messages during translation of a computer program is a necessary condition for reasonable [program] correctness.

The absence of error messages during translation of a computer program is not a sufficient condition for reasonable [program] correctness.

Now rewrite the first statement as an if-then (necessary):

If a computer program does have error messages during translation, then the program does not have reasonable correctness.

Now rewrite the second statement first without the not:

The absence of error messages during translation of a computer program is a sufficient condition for reasonable [program] correctness.

And rewrite as if-then(sufficient):

If a computer program does not have error messages during translation, then the program does have reasonable correctness.

And now take the negation:

There is a computer program that does not have error messages during translation and does not have reasonable correctness.

Now combine them:

If a computer program does have error messages during translation, then the program does not have reasonable correctness, and there is a computer program that does not have error messages during translation but still does not have reasonable correctness.

50. A frequent-flyer club brochure states, "You may select among carriers only if they offer the same lowest fare." Assuming that "only if" has its formal, logical meaning, does this statement guarantee that if two carriers offer the same lowest fare, the customer will be free to choose between them? Explain.

Take the original statement:

"You may select among carriers only if they offer the same lowest fare."

Translate to if-then (only if):

If you select among carriers, then they offer the same lowest fare.

 S \to F 

Original statement.

Or equivalently:

If the carrier does not offer the same lowest fare, then you did not select among carriers.

 \neg F \to \neg S 

The contrapositive.

Now compare to the comparison statement:

"If two carriers offer the same lowest fare, the customer will be free to choose between them."

This is:

 F \to S 

This is the converse statement, and is not logically equivalent to the original.

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Exercise Set 3.3

Page 166

1. Let C be the set of cities in the world, let N be the set of nations in the world, and let P(c, n) be "c is the capital city of n." Determine the truth values of the following statements.

a. P(\text{Tokyo}, \text{Japan})

True.

b. P(\text{Athens}, \text{Egypt})

False.

c. P(\text{Paris}, \text{France})

True.

d. P(\text{Miami}, \text{Brazil})

False.

2. Let G(x, y) be "x^2 > y." Indicate which of the following statements are true and which are false.

a. G(2, 3)

True. 4 > 3.

b. G(1, 1)

False 1 \cancel{>} 1.

c. G(\dfrac{1}{2}, \dfrac{1}{2})

False. \dfrac{1}{4} \cancel{>} \dfrac{1}{2}.

d. G(-2, 2)

True, 4 > 2.

3. The following statement is true: "\forall nonzero number x, \exists a real number y such that xy = 1." For each x given below, find a y to make the predicate "$xy = 1$" true.

a. x = 2

 y = \frac{1}{2} 

b. x = -1

 y = -1 

c. x = \dfrac{3}{4}

 y = \frac{4}{3} 

4. The following statement is true: "\forall real number x, \exists an integer n such that n > x.". For each x given below, find an n to make the predicate n > x true.

a. x = 15.83

 n = 16 

b. x = 10^8

 n = 10^8 + 1 

c. x = 10^{10^{10}}

 n = 10^{10^{10}} + 1 

The statements in exercises 5-8 refer to the Tarski world given in Figure 3.3.1. Explain why each is true.

5. For every circle x there is a square y such that x and y have the same color.

This is saying that for every circle there is at least one square that has the same color:

circle	square(s) with same color
b	h, g
c	j
d	j

As you can see, this is true, every circle has at least one square that has the same color.

6. For every square x there is a circle y such that x and y have different colors and y is above x.

square	circle(s) that have different color and are above the square
e	c, a, b
g	c, a
j	b

As you can see, this statement is true, every square has at least one circle that has a different color and is above it.

7. There is a triangle x such that for every square y, x is above y.

This is saying there is at least one triangle that is above every square. This is true, as the triangle d is above all the squares, e, g, and j.

8. There is a triangle x such that for every circle y, y is above x.

This is saying there is at least one triangle that is below every circle. This is true, both f and i are triangles that are below every circle, b, c, and a.

9. Let D = E = \{-2, -1, 0, 1, 2\}. Explain why the following statements are true.

a. \forall x in D, \exists y such that x + y = 0.

This is true, for every number in \{-2, -1, 0, 1, 2\} there exists at least one corresponding number in \{-2, -1, 0, 1, 2\} such that the sum of those two numbers is 0. This is true. Take the following cases:

 x = -2, y = 2, x + y = 0 

 x = -1, y = 1, x + y = 0 

 x = 0, y = 0, x + y = 0 

 x = 1, y = -1, x + y = 0 

 x = 2, y = -2, x + y = 0 

b. \exists x in D such that \forall y in E, x + y = y.

This is true, within the set \{-2, -1, 0, 1, 2\}, there exists at least one number such that when summed with every number in \{-2, -1, 0, 1, 2}, the result will always be the one number chosen from the second set. Consider the following cases then:

 x = 0, y = -2, x + y = y 

 x = 0, y = -1, x + y = y 

 x = 0, y = 0, x + y = y 

 x = 0, y = 1, x + y = y 

 x = 0, y = 2, x + y = y 

10. This exercise refers to Example 3.3.3. Determine whether each of the following statements is true or false.

a. \forall student S, \exists a dessert D such that S chose D.

For all students, there is at least one desert that that student chose.

This is true, all students chose pie, and Tim chose both pie and cake..

b. \forall student S, \exists a salad T such that S chose T.

For all students, there is at least one salad that that student chose.

This is false, Yuen did not choose a salad.

c. \exists a dessert D such that \forall student S, S chose D.

There exists at least one desert that all students chose.

This is true, all students chose pie.

d. \exists a beverage B such that \forall student D, D chose B.

There exists at least one beverage that all students chose.

This is false, there is no beverage that every student chose.

e. \exists an item I such that \forall student S, S did not choose I.

There exists at least one item that all students did not choose.

This is false, every item was chosen by at least one student.

f. \exists a station Z such that \forall student S, \exists an item I such that S chose I from Z.

There exists at least one station where at least one item was chosen by all students.

This is true, from the Desserts station, the item pie was chosen by all students.

11. Let S be the set of students at your school, let M be the set of movies that have ever been released, and let V(s, m) be "student s has seen movie m." Rewrite each of the following statements without using the symbol \forall, the symbol \exists, or variables.

a. \exists s \in S such that V(s, \text{Casablanca}).

There is at least one student at your school that has seen Casablanca.

b. \forall s \in S, V(s, \text{Star Wars}).

Every student at your school has seen Star Wars.

c. \forall s \in S, \exists m \in M \text{ such that } V(s, m).

Every student at your school as seen at least one movie.

d. \exists m \in M \text{ such that } \forall s \in S, V(s, m).

There is at least one movie that every student at your school as seen.

e. \exists s \in S, \exists t \in S, \text{ and } \exists m \in M \text{ such that } s \neq t \text{ and } V(s, m) \wedge V(t, m).

There are at least two different students at your school that have both seen the same movie.

f. \exists s \in S \text{ and } \exists t \in S \text{ such that } s \neq t \text{ and } \forall m \in M, V(s, m) \to V(t, m).

There are at least two different students at your school where if the first student has seen a movie, then the second student has also seen that movie.

12. Let D = E = \{-2, -1, 0, 1, 2\}. Write negations for each of the following statements and determine which is true, the given statement or its negation.

a. \forall x in D, \exists y such that x + y = 1.

Negation:

\exists x in D, such that \forall y, x + y \neq 1

The negation is true. The original statement cannot be true as there is no corresponding number for x = -2 that would result in x + y = 1. For the negation to be true, there just has to be at least one value for x such that x + y \neq 1, so x = -2, y = -2, x + y = -4 \neq 1 is just one counterexample.

b. \exists x in D such that \forall y in E, x + y = -y.

Negation:

\forall x in D, \exists y in E such that x + y \neq -y.

The negation is true. The original statement cannot be true as there is no single x for which when summed with every number, y from E would equal -y. The negation however states that for every x value in D, there is at least one number y in E where the sum does not equal -y.

c. \forall x in D, \exists y in E such that xy \geq y.

Negation:

\exists x in D, such that \forall y in E, xy < y.

The original statement is true. For the negation to be true there would have to be at least one x in D where every y in E would be greater than the product of xy, and none of the values in E would satisfy this condition.

d. \exists x in D such that \forall y in E, x \leq y.

Negation:

\forall x in D, \exists y in E such that x > y.

The original statement is true. For the negation to be true, every x in D would have to be greater than at least one y value in E, but because D = E, this can never be the case.

In each of 13-19, (a) rewrite the statement in English without using the symbol \forall or \exists or variables and expressing your answer as simply as possible, and (b) write a negation for the statement.

13. \forall color C, \exists an animal A such that A is colored C.

a. For every color, there is an animal of that color.

b. Negation: There is a color that every animal is not that color.

14. \exists a book b such that \forall person p, p has read b.

a. There is a book that every person has read.

b. Negation: For every book, there is a person who has not read that book.

There is no book that every person has read.

15. \forall odd integer n, \exists an integer k such that n = 2k + 1.

a. Given any integer n, there is at least one integer k such that n = 2k + 1.

b. Negation: There exists at least one odd integer n, such that for all integers k, n \neq 2k + 1.

16. \exists a real number u such that \forall real number v, uv = v.

a. There is a real number, u, that when multiplied with any real number, v, uv = v.

b. Negation: For all real numbers, u, there exists at least one real number v, where uv \neq v.

17. \forall r \in \mathbb{Q}, \exists integers a and b such that r = \dfrac{a}{b}.

a. For all rational numbers, r, there exists at least two integers, a, and b, where r = \dfrac{a}{b}.

b. Negation: There exists some rational number r, such that for all integers, a, and b, r \neq \dfrac{a}{b}.

18. \forall x \in \mathbb{R}, \exists a real number y such that x + y = 0.

a. Given any real number, x, there exists a real number y where x + y = 0.

b. Negation: There is a real number x, which when summed with any real number y, x + y \neq 0.

19. \exists x \in \mathbb{R} such that for every real number y, x + y = 0.

a. There is a real number x, that when summed with any real number y, x + y = 0.

b. Negation: Given any real number x, there is a real number y where x + y \neq 0.

20. Recall that reversing the order of the quantifiers in a statement with two different quantifiers may change the truth value of the statement - but it does not necessarily do so. All the statements in the pairs below refer to the Tarski world of Figure 3.3.1. In each pair, the order of the quantifiers is reversed but everything else is the same. For each pair, determine whether the statements have the same or opposite truth values. Justify your answers.

a.

(1) For every square y there is a triangle x such that x and y have different colors.

This is true, consider the table:

square	triangle that has a different color
e	f, i
h	d
g	d
j	d, f, i

As you can see, for every square, there is at least one triangle that has a different color.

(2) There is a triangle x such that for every square y, x, and y have different colors.

This is false, there is no single triangle that is a different color from every square.

b.

(1) For every circle y there is a square x such that x and y have the same color.

For every circle, there is at least one square that has the same color. This is true. Consider the table:

circle	square with the same color as the circle
a	j
b	h, g
c	j

(2) There is a square x such that for every circle y, x and y have the same color.

There is at least one square that has the same color as every circle. This cannot be true unless every circle were the same color and the square was also the same color, but every circle is not the same color so this statement cannot be true.

21. For each of the following equations, determine which of the following statements are true:

(1) For every real number x, there exists a real number y such that the equation is true.

(2) There exists a real number x, such that for every real number y, the equation is true.

Note that it is possible for both statements to be true or for both to be false.

a. 2x + y = 7

Let P(x, y) = 2x + y = 7

(1)

 \forall x \in \mathbb{R}, \exists y \in \mathbb{R}, \text{ such that } P(x, y) 

This is true. For any x, there is always at least one y where y = 7 - 2x.

(2)

 \exists x \in \mathbb{R}, \text{ such that }\forall y \in \mathbb{R}, P(x, y) 

This is false, the statement claims there exists some x which when doubled and added to any y will equal 7. In other words, x = \dfrac{7 - y}{2} for any y, and that is impossible.

b. y + x = x + y

Let P(x, y) = y + x = x + y

(1)

 \forall x \in \mathbb{R}, \exists y \in \mathbb{R}, \text{ such that } P(x, y) 

This is true, as it states that for any real number x, there exists at least one real number y such that their sum, y + x is equal to x + y. By the commutative property of addition, this is always true.

(2)

 \exists x \in \mathbb{R}, \text{ such that }\forall y \in \mathbb{R}, P(x, y) 

This is also true, as it states there exists at least one real number x, where given any real number y, y + x = x + y, which is also true by the commutative property of addition.

c. x^2 - 2xy + y^2 = 0

Let P(x, y) = x^2 - 2xy + y^2 = 0

Notice x^2 - 2xy - y^2 = (x - y)^2 = 0.

(1)

 \forall x \in \mathbb{R}, \exists y \in \mathbb{R}, \text{ such that } P(x, y) 

Yes this is true. This is stating "For any real number x, there is at least one corresponding real number y, such that $x - y = 0$". This is a true statement.

(2)

 \exists x \in \mathbb{R}, \text{ such that }\forall y \in \mathbb{R}, P(x, y) 

This is false, this is stating "There exists some real number x, where for any real number y, $x - y = 0$", and that is impossible, as that would indicate that x is some number universally equal to any other real number.

d. (x - 5)(y - 1) = 0

Let P(x, y) = (x - 5)(y - 1) = 0

(1)

 \forall x \in \mathbb{R}, \exists y \in \mathbb{R}, \text{ such that } P(x, y) 

This is true, no matter what value for x is chosen, there is always at least one real number y, for which P(x, y) is true, namely y = 1.

(2)

 \exists x \in \mathbb{R}, \text{ such that }\forall y \in \mathbb{R}, P(x, y) 

This is true, there is at least one number x such that when multiplied by any value for y - 1 will equal 0, namely x = 5.

e. x^2 + y^2 = -1

Let P(x, y) = x^2 + y^2 = -1

(1)

 \forall x \in \mathbb{R}, \exists y \in \mathbb{R}, \text{ such that } P(x, y) 

This is false because the statement x^2 + y^2 = -1 can never be true, as both x^2 and y^2 must be greater than or equal to 0, and so therefore x^2 + y^2 can never equal a negative value.

(2)

 \exists x \in \mathbb{R}, \text{ such that }\forall y \in \mathbb{R}, P(x, y) 

This is false because the statement x^2 + y^2 = -1 can never be true, as both x^2 and y^2 must be greater than or equal to 0, and so therefore x^2 + y^2 can never equal a negative value.

In 22 and 23, rewrite each statement without using variables or the symbol \forall or \exists. Indicate whether the statement is true or false.

a. \forall real number x, \exists a real number y such that x + y = 0.

Given any real number x, there is a real number y where x + y = 0.

This is true, any real number has an additive inverse.

b. \exists a real number y such that \forall real number x, x + y = 0.

This is false, this is saying there exists a real number y, which when added to any real number x, will equal 0. There is no such number, so this statement is false.

a. \forall nonzero real number r, \exists a real number s such that rs = 1.

This is true, any nonzero real number r will have a reciprocal s such that rs = 1.

b. \exists a real number r such that \forall nonzero real number s, rs = 1.

This is false, this is claiming there is a real number r that is the reciprocal for any real number s. There is no such number, so this statement is false.

24. Use the laws for negating universal and existential statements to derive the following rules:

a. \neg(\forall x \in D(\forall y \in E(P(x, y)))) \equiv \exists x \in D(\exists y \in E(\neg P(x, y)))

 \neg(\forall x \in D(\forall y \in E(P(x, y)))) \equiv \exists x \in D(\neg(\forall y \in E(P(x, y)))) 

 \quad \equiv \exists x \in D(\exists y \in E \neg P(x, y)) 

b. \neg(\exists x \in D(\exists y \in E(P(x, y)))) \equiv \forall x \in D(\forall y \in E(\neg P(x, y)))

 \neg(\exists x \in D(\exists y \in E(P(x, y)))) \equiv \forall x \in D \neg(\exists y \in E(P(x, y))) 

 \quad \equiv \forall x \in D(\forall y \in E(\neg P(x, y))) 

Each statement in 25-28 refers to the Tarski world of Figure 3.3.1. For each, (a) determine whether the statement is true or false and justify your answer, and (b) write a negation for the statement (referring, if you wish, to the result in exercise 24).

25. \forall circle x and \forall square y, x is above y.

(a) This is true, all circles are above all squares.

(b) \exists circle x and \exists square y such that x is not above y.

26. \forall circle x and \forall triangle y, x is above y.

(a) This is false, as not all circles are above all triangles (circles b and c are not above triangle d).

(b) \exists circle x and \exists triangle y such that x is not above y.

27. \exists a circle x and \exists a square y such that x is above y and x and y have different colors.

(a) This is true, there exists at least one circle and at least one square where the circle is above the square and the circle and square have different colors (all circles, a, b, and c are above all squares, and circle b has a different color from squares e, and j; circle c has a different color from squares e and g; and circle a has different colors from squares e and g).

(b) \forall circle x and \forall square y, x is not above y and x and y are the same color.

28. \exists a triangle x and \exists a square y such that x is above y and x and y have the same color.

(a) This is true, this is saying there exists at least one triangle and at least one square where the triangle is above the square and the triangle and the square are the same color. Triangle d fulfills both conditions, as it is above all squares, and is the same color as square e.

(b) \forall triangles x and \forall squares y, x is not above y and x and y are not the same color.

For each of the statements in 29 and 30, (a) write a new statement by interchanging the symbols \forall and \exists, and (b) state which is true: the given statement, the version with interchanged quantifiers, neither or both.

29. \forall x \in \mathbb{R}, \exists y \in \mathhbb{R} such that x < y.

(a) \exists x \in \mathbb{R}, \forall y \in \mathbb{R} such that x < y.

(b) The original statement is true (there always exists some real number y where y = x + 1, such that x < y). The interchanged statement is false, as there is no single real number x that is less than all real numbers y.

30. \exists x \in \mathbb{R} such that \forall y \in \mathbb{R}^{-} (the set of negative real numbers), x > y.

(a) \forall x \in \mathbb{R}, \exists y \in \mathbb{R}^{-} such that x > y.

(b) The original statement is true, as it is saying that there exists some real number that is greater than any negative real number (any positive number or 0 would fulfill this condition.) The interchanged statement is true, as for any real number x, you can always choose some negative number y such that y = x - 1 where x > y is true.

31. Consider the statement "Everybody is older than somebody." Rewrite this statement in the form "\forall people x, \exists ______."

"\forall people x, \exists some person y, such that x is older than y."

32. Consider the statement "Somebody is older than everybody." Rewrite this statement in the form "\exists a person x such that \forall ______."

"\exists a person x such that \forall people y, x is older than y."

In 33-39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement.

33. Everybody loves somebody.

(a) "\forall people x, \exists some person y such that x loves y."

(b) Negation: "\exists some person x such that \forall people y, x does not love y."

34. Somebody loves everybody.

(a) "\exists some person x such that \forall people y, x loves y."

(b) Negation: "\forall people x, \exists some person y, such that x does not love y."

35. Everybody trusts somebody.

(a) "\forall people x, \exists some person y such that x trusts y."

(b) Negation: "\exists some person x, such that \forall people y, x does not trust y."

36. Somebody trusts everybody.

(a) "\exists some person x such that \forall people y, x trusts y."

(b) Negation: "\forall people x, \exists some person y, such that x does not trust y."

37. Any even integer equals twice some integer.

(a) "\forall even integer n, \exists some integer k such that n = 2k."

(b) Negation: "\exists some integer n, such that \forall integers k, n \neq 2k."

38. Every action has an equal and opposite reaction.

(a) "\forall actions x, \exists some action y, such that y is the equal and opposite reaction to x."

(b) Negation: "\exists some action x such that \forall actions y, y is not the equal and opposite reaction to x."

39. There is a program that gives the correct answer to every question that is posed to it.

(a) "\exists some program x such that \forall questions y, x gives the correct answer to y."

(b) Negation: "\forall programs x, \exists some question y such that x does not give the correct answer to y."

40. In informal speech most sentences of the form "There is ______ every ______" are intended to be understood as meaning "\forall ______ \exists ______," even though the existential quantifier there is comes before the universal quantifier every. Note that this interpretation applies to the following well-known sentences. Rewrite them using quantifiers and variables.

a. There is a sucker born every minute.

"\forall minutes x, \exists some sucker y such that y is born in minute x."

b. There is a time for every purpose under heaven.

"\forall purpose under heaven x, \exists some time y such that y is the time for x."

41. Indicate which of the following statements are true and which are false. Justify your answers as best you can.

a. \forall x \in \mathbb{Z}^{+}, \exists y \in \mathbb{Z}^{+} such that x = y + 1.

This is false, this is stating that given any positive integer x, there exists some positive integer y that such that y = x - 1. But take x = 1, and 1 = y + 1 means y = 0, and that means this statement is false when x = 1.

b. \forall x \in \mathbb{Z}, \exists y \in \mathbb{Z} such that x = y + 1.

This is statement is saying that given any integer x, there exists at least one integer y, such that x = y + 1. This is true. Take any y = x - 1, and you will still have an integer.

c. \exists x \in \mathbb{R} such that \forall y \in \mathbb{R}, x = y + 1 .

This statement is saying there exists some real number x that is equal to any real number y plus 1. This is false. You can't have some universal number x that is always 1 greater than all possible real numbers, $y.

d. \forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+} such that xy = 1.

This statement is saying that given any positive real number, x, there is at least one positive real number y, such that xy = 1. This is true, as for any positive real number, there is always a reciprocal.

e. \forall x \in \mathbb{R}, \exists y \in \mathbb{R} such that xy = 1.

This statement is saying that given any real number, x, there is at least one real number y, such that xy = 1. This statement is false. If x = 0, then there is no y such that xy = 1$.

f. \exists x \in \mathbb{R} such that \forall y \in \mathbb{R}, x + y = y.

This statement is saying there is at least one real number x such that given any real number y, x + y = y. This is true, there is a real number for x, specifically x = 0, where x plus any real number y will equal y.

g. \forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+} such that y < x.

This statement is saying that given any positive real number x, there is at least one positive real number y such that y < x. This is true. This is saying that for any positive real number, there is always a positive real number that is less than it.

h. \exists x \in \mathbb{R}^{+} such that \forall y \in \mathbb{R}^{+}, x \leq y.

This statement is saying there exists some positive real number x such that given any positive real number y, x \leq y, or some universal x will always be less than or equal to any y. This is false, since it is possible that y = \dfrac{x}{2}, there cannot be some universal x in the set of positive real numbers that is always less than all positive real numbers y.

42. Write the negation of the definition of limit of a sequence given in Example 3.3.7.

Definition of a limit:

\forall \varepsilon > 0, \exists N \in \mathbb{Z}, \forall n \in \mathbb{Z}((n > N) \to (L - \varepsilon < a_n < L + \varepsilon))

Negation:

\exists \varepsilon > 0, \forall N \in \mathbb{Z}, \exists n \in \mathbb{Z} ((n > N) \wedge ((L - \varepsilon \geq a_n) \vee (a_n \geq L + \varepsilon)))

43. The following is the definition for \lim\limits_{x \to a}f(x) = L:

For every real number \varepsilon > 0, there exists a real number \delta > 0 such that for every real number x, if a - \delta < x < a + \delta and x \neq a then

 L - \varepsilon < f(x) < L + \varepsilon 

Write what it means for \lim\limits_{x \to a}f(x) \neq L. In other words, write the negation of the definition.

Original:

 \lim\limits_{x \to a}f(x) = L 

 \forall \varepsilon \in \mathbb{R} > 0, \exists \delta \in \mathbb{R} > 0, \forall x \in \mathbb{R} (((a - \delta < x < a + \delta) \wedge (x \neq a)) \to L - \varepsilon < f(x) < L + \varepsilon) 

Negation:

 \exists \varepsilon \in \mathbb{R} > 0, \forall \delta \in \mathbb{R} > 0, \exists x \in \mathbb{R} ((a - \delta < x < a + \delta) \wedge (x \neq a)) \wedge ((L - \varepsilon \geq f(x)) \vee (f(x) \geq L + \varepsilon)))

There exists some real number \varepsilon > 0 such that given any real number \delta > 0, there exists some real number x such that (a - \delta < x < a + \delta) and x \neq a and L - \varepsilon \geq f(x) or f(x) \geq L + \varepsilon is true.

44. The notation \exists ! stands for the words "there exists a unique." Thus, for instance, "\exists ! x such that x is prime and x is even" means that there is one and only one even prime number. Which of the following statements are true and which are false?

a. \exists ! real number x such that \forall real number y, xy = y.

The statement is true, as there exists a real number x for which given any real number y, xy = y. That number is x = 1, and only x = 1 satisfies this condition (satisfying the unique constraint).

b. \exists ! integer x such that \dfrac{1}{x} is an integer.

The statement is false , there exists two integers for x such that \dfrac{1}{x} is an integer and those are x = 1 and x = -1, violating the unique constraint.

c. \forall real number x, \exists ! real number y such that x + y = 0.

The statement is true, given any real number x, there exists a single unique number that is it's additive inverse, namely it's corresponding negative number.

45. Suppose that P(x) is a predicate and D is the domain of x. Rewrite the statement "$\exists ! x \in D \text{ such that } P(x)$" without using the symbol \exists !. (See exercise 44 for the meaning of \exists !.)

Original:

\exists ! x \in D, P(x)

Alternative:

 \exists x \in D (P(x) \to \forall y \in D(P(y) \to (x = y))) 

In 46-54, refer to the Tarski world given in Figure 3.1.1, which is shown again here for reference. The domains of all variables consist of all the objects in the Tarski world. For each statement, (a) indicate whether the statement is true or false and justify your answer, (b) write the given statement using the formal logical notation illustrated in Example 3.3.10, and (c) write a negation for the given statement using the formal logical notation of Example 3.3.10.

46. There is a triangle x such that for every square y, x is above y.

(a) This is true, there is at least one triangle that is above every square, both triangles a and c are cases of this.

(b) \exists a triangle x such that \forall squares y, x is above y.

c \forall triangles x, \exists some square y such that x is not above y.

47. There is a triangle x such that for every circle y, x is above y.

(a) This is false, there is no triangle that is above every circle.

(b) \exists a triangle x such that \forall circles y, x is above y.

c \forall triangles x, \exists a circle y such that x is not above y.

48. For every circle x, there is a square y such that y is to the right of x.

(a) This is false, there is a circle that is not to the right of a square, and that is circle b.

(b) \forall circles x, \exists a square y such that y is to the right of x.

c \exists a circle x such that \forall squares y, y is not to the right of x.

49. For every object x, if x is a circle then there is a square y such that y has the same color as x.

(a) This is false, there is a circle that does not have the same color as at least one square, and those circles are d, f, i, and k.

(b)

Let P(x) be "x is a circle" S(y) be "y is a square" and Q(x, y) be "y has the same color as x."

\forall objects x, (P(x) \to \exists \text{ some square } y (S(y) \wedge Q(x, y)))

c

Let P(x) be "x is a circle" S(y) be "y is a square" and Q(x, y) be "y has the same color as x."

\exists some object x (P(x) \wedge \forall \text{ squares } y (\neg S(y) \vee \neg Q(x, y)))

50. For every object x, if x is a triangle then there is a square y such that y is below x.

(a) This is true, for every triangle, there is at least one square that is below it.

(b)

Let P(x) be "x is a triangle", S(y) be "y is a square", and Q(x, y) be "y is below $x$".

\forall objects x, (P(x) \to \exists \text{ some square } y (S(y) \wedge Q(x,y)))

c

\exists some object x such that P(x) \wedge \forall y (\neg S(y) \vee \neg Q(x, y))

Let P(x) be "x is a triangle", S(y) be "y is a square", and Q(x, y) be "y is below $x$".

51. There is a square x such that for every triangle y, if y is above x then y has the same color as x.

(a) This is true, there is at least one square where every triangle above it is the same color, this square is e with triangles c and a above it being the same color of blue.

(b)

Let be "S(x)" be "x is a square ", and T(y) be "y is a triangle", and A(x, y) be "y is above $x$" and C(x, y) be "y has the same color as x."

\exists some x such that (S(x) \wedge ((T(y) \wedge A(x, y)) \to C(x, y)))

c

\forall objects x, \neg S(x) \vee \exists \text{ some } y ((T(y) \wedge A(x, y)) \wedge \neg C(x, y))

Let be "T(x)" be "x is a triangle", and S(y) be "y is a square", and A(x, y) be "y is above $x$" and C(x, y) be "y has the same color as x."

52. For every circle x and for every triangle y, x is to the right of y.

(a) This is false, every circle is not to the right of every triangle. For example circle b is not to the right of triangle c.

(b)

Let C(x) be "x is a circle", let T(y) be "y is a triangle", let R(x, y) be "x is to the right of y."

\forall x \forall y ((C(x) \wedge T(y)) \to R(x, y))

c

\exists x \exists y (C(x) \wedge T(y) \wedge \neg R(x, y))

53. There is a circle x and there is a square y such that x and y have the same color.

(a) This is true, there is at least one circle and at least one square that have the same color, that is b and (h or j).

(b)

Let C(x) be "x is a circle", and T(y) be "y is a triangle", and C(x, y) be "x and y have the same color."

 \exists x \exists y (C(x) \wedge T(y) \wedge C(x, y)) 

c

 \forall x \forall y (\neg C(x) \vee \neg T(y) \vee \neg C(x, y)) 

Let C(x) be "x is a circle", and T(y) be "y is a triangle", and C(x, y) be "x and y have the same color."

54. There is a circle x and there is a triangle y such that x has the same color as y.

(a) This is false, there is not any single circle that has the same color as any single triangle.

(b)

Let C(x) be "x is a circle", and T(y) be "y is a triangle", and C(x, y) be "x has the same color as y."

 \exists x \exists y (C(x) \wedge T(y) \wedge C(x, y)) 

c

Let C(x) be "x is a circle", and T(y) be "y is a triangle", and C(x, y) be "x has the same color as y."

 \forall x \forall y (\neg C(x) \vee \neg T(y) \vee \neg C(x, y)) 

Let P(x) and Q(x) be predicates and suppose D is the domain of x. In 55-58, for the statement forms in each pair, determine whether (a) they have the same truth value for every choice of P(x), Q(x) and D, or (b) there is a choice of P(x), Q(x), and D for which they have opposite truth values.

55. \forall x \in D, (P(x) \wedge Q(x)) \text{ and } (\forall x \in D, P(x)) \wedge (\forall x \in D, Q(x))

These two statements are equivalent by the distributive law.

56. \exists x \in D, (P(x) \wedge Q(x)) \text{ and } (\exists x \in D, P(x)) \wedge (\exists x \in D, Q(x))

These two statements are not equivalent. When we separate the statement into two existential quantifiers on the right side of the equivalency, we are saying "there exists some x in D for which $P(x) and there exists some x in D for which Q(x) is true." But we could choose a different x for P(x) and a different x for Q(x), which is not what is happening on the left side of the equivalency, where the same x is being chosen for both P(x) and Q(x).

57. \forall x \in D, (P(x) \vee Q(x)) \text{ and } (\forall x \in D, P(x)) \vee (\forall x \in D, Q(x))

These two statements are not equivalent. On the left side of the equivalency, we say that for any given x, either P(x) or Q(x) can be true. On the right side of the equivalency, we say that for any given x, P(x) must be true or for any given x, Q(x) must be true, but again, the two $x$'s do not have to be the same for these two predicates and so the right side equivalency is essentially stricter saying that either P(x) must be true for every x or Q(x) must be true for every x. This is as opposed to the left side which says for any given x, either P(x) or Q(x) can be true.

58. \exists x \in D, (P(x) \vee Q(x)) \text{ and } (\exists x \in D, P(x)) \vee (\exists x \in D, Q(x))

These are equivalent. The left side is saying that there is an x in D, where either P(x) or Q(x) or both can be true. The right side is saying there is an x in D, such that P(x) can be true or there is an x in D such that Q(x) can be true. These are equivalent statements.

In 59-61, find the answers Prolog would give if the following questions were added to the program given in Example 3.3.11.

a. ?\text{isabove}(b_1, w_1)

True.

b. ?\text{color}(X, \text{white})

X = w_2, X = w_1

c. ?\text{isabove}(X, b_3)

X = b_2, X = w_2

a. ?\text{isabove}(w_1, g)

False.

b. ?\text{color}(w_2, \text{blue})

False.

c. ?\text{isabove}(X, b_1)

X = g

a. ?\text{isabove}(w_2, b_3)

True.

b. ?\text{color}(X, \text{gray})

X = g

c. ?\text{isabove}(g, X)

X = b_1, X = w_1

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Exercise Set 3.4

Page 179

1. Let the following law of algebra be the first statement of an argument: For all real numbers a and b,

 (a + b)^2 = a^2 + 2ab + b^2 

Suppose each of the following statements is, in turn, the second statement of the argument. Use universal instantiation or universal modus ponens to write the conclusion that follows in each case.

a. a = x and b = y are particular real numbers.

\therefore (x + y)^2 = x^2 + 2xy + y^2

b. a = f_i and b = f_j are particular real numbers.

\therefore (f_i + f_j)^2 = f_i^2 + 2f_if_j + f_j^2

c. a = 3u and b = 5v are particular real numbers.

\therefore (3u + 5v)^2 = (3u)^2 + 2(3u)(5v) + (5v)^2

d. a = g(r) and b = g(s) are particular real numbers.

\therefore (g(r) + g(s))^2 = (g(r))^2 + 2(g(r))(g(s)) + (g(s))^2

e. a = \log(t_1) and b = \log(t_2) are particular real numbers.

\therefore (\log(t_1) + \log(t_2))^2 = (\log(t_1))^2 + 2(\log(t_1))(\log(t_2)) + (\log(t_2))^2

Use universal instantiation or universal modus ponens to fill in valid conclusions for the arguments in 2-4.

\text{If an integer} n \text{ equals } 2 \cdot k \text{ and } k \text{ is an integer, then } n \text{ is even.} \ 0 \text{ equals } 2 \cdot 0 \text{ and } 0 \text{ is an integer.} \ \therefore \text{______}

\therefore 0 \text{ is even.}

\text{For all real numbers } a, b, c, \text{ and } d, \text{ if } b \neq 0 \text{ and } d \neq 0 \text{ then } \frac{a}{b} + \frac{c}{d} = \frac{(ad + bc)}{bd} \ a = 2, b = 3, c = 4, \text{ and } d = 5 \text{ are particular real numbers such that } b \neq 0 \text{ and } d \neq 0 \ \therefore \text{______}

\therefore \dfrac{2}{3} + \dfrac{4}{5} = \dfrac{((2)(5) + (3)(4))}{(3)(5)} = \dfrac{22}{15}

\forall \text{ real numbers } r, a, \text{ and } b, \text{ if } r \text{ is positive, then } (r^q)^b = r^{ab} \ r = 3, a = \frac{1}{2}, \text{ and } b = 6 \text{ are particular real numbers such that } r \text{ is positive.} \ \therefore \text{______}

\therefore (3^{\frac{1}{2}})^6 = r^{(\frac{1}{2})(6)} = r^3

Use universal modus tollens to fill in valid conclusions for the arguments in 5 and 6.

\text{All irrational numbers are real numbers.} \ \frac{1}{0} \text{ is not a real number.} \ \therefore \text{______}

\therefore \dfrac{1}{0} \text{ is not an irrational number.}

\text{If a computer program is correct, then compilation of the program does not produce error messages.} \ \text{Compilation of this program produces error messages.} \ \therefore \text{______}

\therefore \text{ this program is not correct.}

Some of the arguments in 7-18 are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers.

\text{All healthy people eat an apple a day.} \ \text{Keisha eats an apple a day.} \ \therefore \text{Keisha is a healthy person.}

This is invalid, this is an example of a converse error.

\text{All freshmen must take a writing course.} \ \text{Caroline is a freshman.} \ \therefore \text{Caroline must take a writing course.}

This is valid, an example of universal modus ponens.

\text{If a graph has no edges, then it has a vertex of degree zero.} \ \text{This graph has at least one edge.} \ \therefore \text{This graph does not have a vertex of degree zero.}

Invalid, inverse error.

\text{If a product of two numbers is } 0 \text{, then at least one of the numbers is } 0 \text{.} \ \text{For a particular number } x \text{, neither } (2x + 1) \text{ nor } (x - 7) \text{ equals } 0 \text{.} \ \therefore \text{The product } (2x + 1)(x - 7) \text{ is not } 0 \text{.}

This is valid, an example of universal modus tollens.

\text{All cheaters sit in the back row.} \ \text{Monty sits in the back row.} \ \therefore \text{Monty is a cheater.}

Invalid, converse error.

\text{If an 8-bit two's complement represents a positive integer, then the 8-bit two's complement starts with a 0.} \ \text{The 8-bit two's complement for this integer does not start with 0.} \ \therefore \text{This integer is not positive.}

Valid, universal modus tollens.

\text{For every student } x \text{, if } x \text{ studies discrete mathematics, then } x \text{ is good at logic.} \ \text{Tarik studies discrete mathematics.} \ \therefore \text{Tarik is good at logic.}

Valid, universal modus ponens.

\text{If compilation of a computer program produces error messages, then the program is not correct.} \ \text{Compilation of this program does not produce error messages.} \ \therefore \text{This program is correct.}

Invalid, inverse error.

\text{Any sum of two rational numbers is irrational.} \ \text{The sum } r + s \text{ is rational.} \ \therefore \text{The numbers } r \text{ and } s \text{ are both rational.}

Valid, universal modus ponens.

\text{If a number is even, then twice that number is even.} \ \text{The number } 2n \text{ is even, for a particular number } n \text{.} \ \therefore \text{The particular number } n \text{ is even.}

Invalid, converse error.

\text{If an infinite series converges, then the terms go to 0.} \ \text{The terms of the infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ go to 0.} \ \therefore \text{The infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ converges.}

Invalid, converse error.

\text{If an infinite series converges, then the terms go to 0.} \ \text{The terms of the infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ do not go to 0.} \ \therefore \text{The infinite series } \sum_{n=1}^{\infty}{\frac{1}{n}} \text{ does not converge.}

Valid, by universal modus tollens.

19. Rewrite the statement "No good cars are cheap" in the form "\forall x if P(x) then \neg Q(x)." Indicate whether each of the following arguments is valid or invalid, and justify your answers.

a.

\text{No good car is cheap.} \ \text{A Rimbaud is a good car.} \ \therefore \text{A Rimbaud is not cheap.}

\forall x (P(x) \to \neg Q(x)) \ P(a) \ \therefore \neg Q(a)

This is valid by universal modus ponens.

b.

\text{No good car is cheap.} \ \text{A Simbaru is not cheap.} \ \therefore \text{A Simbaru is a good car.}

\forall x (P(x) \to \neg Q(x)) \ \neg Q(a) \ \therefore \neg P(a)

This is invalid, converse error.

c.

\text{No good car is cheap.} \ \text{A VX Roadster is cheap.} \ \therefore \text{A VX Roadster is not good.}

\forall x (P(x) \to \neg Q(x)) \ \Q(a) \ \therefore \neg P(a)

This is true by universal modus tollens.

d.

\text{No good car is cheap.} \ \text{An Omnex is not a good car.} \ \therefore \text{An Omnex is cheap.}

\forall x (P(x) \to \neg Q(x)) \ \neg P(a) \ \therefore Q(a)

Invalid, inverse error.

a. Use a diagram to show that the following argument can have true premises and a false conclusion.

\text{All dogs are carnivorous.} \ \text{Aaron is not a dog.} \ \therefore \text{Aaron is not carnivorous.}

Our diagram shows two discs, one labeled carnivorous, the other labeled dogs, dogs lies completely inside carnivorous. We have labeled two points, both "Aaron", one lies outside both carnivorous and dogs, the other lies within carnivorous and not dogs, showing that we have two true premises, but a false conclusion. This is an example of an inverse error.

b. What can you conclude about the validity or invalidity of the following argument form? Explain how the result from part (a) leads to this conclusion.

\forall x, \text{ if } P(x) \text{ then } Q(x) \ \neg P(a) \text{ for a particular } a \ \therefore \neg Q(a)

This is an example of an inverse error. Just because we know \neg P(a) does not mean we can conclude anything about Q(a).

Indicate whether the arguments in 21-27 are valid or invalid. Support your answers by drawing diagrams.

\text{All people are mice.} \ \text{All mice are mortal.} \ \therefore \text{All people are mortal.}

This argument is valid, even though one of the premises is false (all people are mice). Drawn out this is simply three discs, one being all mortal, second being all mice, and third being all people, people inside of mice, mice inside of mortal. The conclusion is true, this is an example of universal transitivity.

\text{All discrete mathematics students can tell a valid argument from an invalid one.} \ \text{All thoughtful people can tell a valid argument from an invalid one.} \ \therefore \text{All discrete mathematics students are thoughtful.}

Let D(x) be "x is a discrete math student."

Let P(x) be "x is a thoughtful person."

Let Q(x) be "x can tell a valid argument from an invalid one."

Our argument then can be read as:

\forall x (D(x) \to Q(x)) \ \forall x (P(x) \to Q(x)) \ \therefore \forall x (D(x) \to P(x))

This is an invalid syllogism. In diagrams, one can see that while D(x) \in Q(x) and P(x) \in Q(x), we do not know if D(x) or P(x) overlap or not, we only know that both of them lie within Q(x). Therefore this argument is invalid.

\text{All teachers occasionally make mistakes} \ \text{No gods ever make mistakes.} \ \therefore \text{No teachers are gods.}

This argument is valid. This is an example of universal modus tollens. In diagrams, we have a disc representing all people who make mistakes, inside we have a disc labeled teachers, and outside of both discs we have a separate disc labeled gods. This diagram also illustrates that if one is a teacher, which is within the set of people that make mistakes, then a teacher cannot also be in the set of gods, which explicitly lies outside the set of people that make mistakes.

\text{No vegetarians eat meat.} \ \text{All vegans are vegetarians.} \ \therefore \text{No vegans eat meat.}

This argument is valid. This is an example of universal transitivity. In diagrams, we have two separated discs, one labeled "vegetarians", the other labeled "eats meat". Inside the "vegetarians" disc we have a smaller disc labeled "vegans." We can see therefore that no vegans eat meat.

\text{No college cafeteria food is good.} \ \text{No good food is wasted.} \ \therefore \text{No college cafeteria food is wasted.}

This argument is invalid.

Let P(x) be "x is college cafeteria food."

Let Q(x) be "x is good food."

Let R(x) be "x food is wasted."

Our argument formally looks like:

\forall x (P(x) \to \neg Q(x)) \ \forall x (Q(x) \to \neg R(x)) \ \therefore \forall x (P(x) \to \neg R(x))

But we established that no college cafeteria food is good, so it does not follow that if the food is good, then it is not wasted means that if the food is cafeteria food, then it is not wasted.

In diagrams, we have two separate discs, P(x) and Q(x), and inside Q(x) there is a third disc labeled \neg R. We can see therefore that just because x is cafeteria food, that does not mean that x is not wasted.

\text{All polynomial functions are differentiable.} \ \text{All differentiable functions are continuous.} \ \therefore \text{All polynomial functions are continuous.}

This argument is valid. It is an example of universal transitivity. In a diagram, there is a disc labeled polynomial functions, there is a disc inside that disc labeled "differentiable" and inside "differentiable" there is a third disc labeled "continuous." We can see then that the set of all polynomial functions contain the set of all continuous functions.

[Adapted from Lewis Carrol.]

\text{Nothing intelligible ever puzzles me.} \ \text{Logic puzzles me.} \ \therefore \text{Logic is unintelligible.}

Let P(x) be "x is intelligible".

Let Q(x) be "x puzzles me."

Let a be logic.

Our argument then takes the form:

\forall x (P(x) \to \neg Q(x)) \ Q(a) \ \therefore \neg P(a)

This is a valid argument. It is valid by universal modus tollens. In a diagram, we have two separate discs, one labeled "intelligible", the other labeled "puzzles me". Inside the disc "puzzles me", we have a single point labeled "Logic." We can see that the point of Logic does not lie inside of "intelligible", so therefore we can reasonably conclude that Logic is unintelligible.

In exercises 28-32, reorder the premises in each of the arguments to show that the conclusion follows as valid consequence from the premises. It may be helpful to rewrite the statements in if-then form and replace some of them by their contrapositives. Exercises 28-30 refer to the kinds of Tarski words discussed in Examples 3.1.13 and 3.3.1. Exercises 31 and 32 are adapted from Symbolic Logic by Lewis Carroll.

1. Every object that is to the right of all the blue objects is above all the triangles.

2. If an object is a circle, then it is to the right of all the blue objects.

3. If an object is not a circle, then it is not gray.

\therefore All the gray objects are above all the triangles.

Answer:

3. If an object is not a circle, then it is not gray.

Restated contrapositive:

3.c If an object is gray, then it is a circle.

2. If an object is a circle, then it is to the right of all the blue objects.

1. If an object is to the right of all the blue objects, then it is above all the triangles.

\therefore If an object is gray, then it is above all the triangles.

1. All the objects that are to the right of all the triangles are above all the circles.

2. If an object is not above all the black objects, then it is not a square.

3. All the objects that are above all the black objects are to the right of all the triangles.

\therefore All the squares are above all the circles.

Answer:

2. If an object is not above all the black objects, then it is not a square.

Rewritten as contrapositive:

2.c If an object is a square, then it is above all black objects.

3. If an object is above all the black objects, then it is to the right of all the triangles.

1. If an object is to the right of all the triangles, then it is above all the circles.

\therefore If an object is a square, then it is above all the circles.

1. If an object is above all the triangles, then it is above all the blue objects.

2. If an object is not above all the gray objects, then it is not a square.

3. Every black object is a square.

4. Every object that is above all the gray objects is above all the triangles.

\therefore If an object is black, then it is above all the blue objects.

Answer:

3. If an object is a black object, then it is a square.

2.c If an object is a square, then it is above all the gray objects.

4. If an object is above all the gray objects, then it is above all the triangles.

1. If an object is above all the triangles, then it is above all the blue objects.

\therefore If an object is black, then it is above all the blue objects.

1. I trust every animal that belongs to me.

2. Dogs gnaw bones.

3. I admit no animals into my study unless they will beg when told to do so.

4. All the animals in the yard are mine.

5. I admit every animal that I trust into my study.

6. The only animals that are really willing to beg when told to do so are dogs.

\therefore All the animals in the yard gnaw bones.

Answer:

4. If the animal is in the yard, then it is mine.

1. If an animal is mine, then I trust the animal.

5. If I trust the animal, then I admit it into my study.

3. If I admit the animal into my study, then it is willing to beg.

6. If the animal is willing to beg, then the animal is a dog.

2. If the animal is a dog, then it gnaws bones.

\therefore If the animal is in the yard, then it gnaws bones.

1. When I work a logic example without grumbling, you may be sure it is one I understand.

2. The arguments in these examples are not arranged in regular order like the ones I am used to.

3. No easy examples make my head ache.

4. I can't understand examples if the arguments are not arranged in regular order like the ones I am used to.

5. I never grumble at an example unless it gives me a headache.

\therefore These examples are not easy.

Answer:

2. The arguments in these examples are not arranged in regular order like the ones I am used to.

4. If the arguments in these examples are not arranged in regular order like the ones I am used to, then the example is one I don't understand.

1c. If an example is one I don't understand, then I grumble.

5. If I do grumble, then an example makes my head ache.

3. If an example makes my head ache, then the example is not easy.

\therefore If it is an example from an argument that is not arranged in regular order like the ones I am used to, then the example is not easy.

In 33 and 34 a single conclusion follows when all the given premises are taken into consideration, but it is difficult to see because the premises are jumbled up. Reorder the premises to make it clear that a conclusion follows logically, and state the valid conclusion that can be drawn. (It may be helpful to rewrite some of the statements in if-then form and to replace some statements by their contrapositives.)

1. No birds except ostriches are at least 9 feet tall.

2. There are no birds in this aviary that belong to anyone but me.

3. No ostrich lives on mince pies.

4. I have no birds less than 9 feet high.

Answer:

2. If a bird is in this aviary, then it belongs only to me.

4. If the bird belongs to me, then it is at least 9 feet high.

1c. If the bird is more than or equal to 9 feet high, then the bird is an ostrich.

3. If a bird is an ostrich, then it does not live on mince pies.

\therefore If a bird is in this aviary, then it does not live on mince pies.

1. All writers who understand human nature are clever.

2. No one is a true poet unless he can stir the human heart.

3. Shakespeare wrote Hamlet.

4. No writer who does not understand human nature can stir the human heart.

5. None but a true poet could have written Hamlet.

Answer:

3. If one is Shakespeare, then one wrote Hamlet.

5c. If one wrote Hamlet, then one is a true poet.

2c. If one is a true poet, then one can stir the human heart.

4c. If one can stir the human heart, then one is a writer who understands human nature.

1. If one is a writer who understands human nature, then one is clever.

\therefore If one is Shakespeare, then one is clever.

35. Derive the validity of universal modus tollens from the validity of universal instantiation and modus tollens.

Universal modus tollens is valid because it is the composition:

 \forall x (P(x) \to Q(x)) \Rightarrow P(a) \to Q(a) \Rightarrow \neg Q(a) \Rightarrow \neg P(a) 

This is just Universal Instantiation and Universal Modus Tollens combined.

36. Derive the validity of universal form of part (a) of the elimination rule from the validity of universal instantiation and the valid argument called elimination in Section 2.3.

 \forall x (P(x) \vee Q(x)) \Rightarrow P(a) \vee Q(a) \Rightarrow Q(a) \text{ given } \neg P(a)