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.

b. All discrete mathematics students are nonathletic.

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

d. No discrete mathematics students are athletic.

e. Some discrete mathematics students are nonathletic.

f. No athletic people are discrete mathematics students.

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.

b. No dogs are loyal.

c. Some dogs are disloyal.

d. Some dogs are loyal.

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

f. There is a dog that is disloyal.

g. No animals that are not dogs are loyal.

h. Some animals that are not dogs are loyal.

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

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

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

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

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

b. All graphs are connected.

c. Some suspicions were substantiated.

d. Some estimates are accurate.

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

a. Every valid argument has a true conclusion.

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

Write a negation for each statement in 6 and 7.

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

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

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

b. This number is not related to any 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.

Write a negation for each statement in 9 and 10.

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

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

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.

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.

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.

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.

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.

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

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

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.

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.

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

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

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

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

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

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

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

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

23. If a function is differentiable then it is 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.

All the females in Tom's family are children.

b.

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

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

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.

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

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

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

27. Exercise 17

28. Exercise 18

29. Exercise 19

30. Exercise 20

31. Exercise 21

32. Exercise 22

33. Exercise 23

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.)

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

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

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.

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.

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?)

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

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.

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

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

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

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

44. A polygon is square only if 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.

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

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

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

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.