diff --git a/chapter_3/exercises.md b/chapter_3/exercises.md index aa87a54..ef3eac0 100644 --- a/chapter_3/exercises.md +++ b/chapter_3/exercises.md @@ -671,3 +671,284 @@ 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. + +--- + +**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. + +6. + +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. + +7. + +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. + +11. + +_Statement:_ The sum of any two irrational numbers is irrational. + +_Proposed negation:_ The sum of any two irrational numbers is rational. + +12. + +_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. + +13. + +_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. + +14. + +_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. diff --git a/chapter_3/notes.md b/chapter_3/notes.md index 0dacfd8..b0a4bf6 100644 --- a/chapter_3/notes.md +++ b/chapter_3/notes.md @@ -58,3 +58,84 @@ Let $P(x)$ and $Q(x)$ be predicates and suppose the domain of $x$ is $D$. - The notation $P(x) \Leftrightarrow Q(x)$ means that $P(x)$ and $Q(x)$ have identical truth sets, or, equivalently, $\forall x, P(x) \leftrightarrow Q(x)$. + +--- + +Page 145 + +**Theorem 3.2.1 Negation of a Universal Statement** + +The negation of a statement of the form + +$$ \forall \text{ in } D, Q(x) $$ + +is logically equivalent to a statement of the form + +$$ \exists \text{ in } D \text{ such that } \neg Q(x) $$ + +Symbolically, + +$$ \neg(\forall x \in D, Q(x)) \equiv \exists x \in D \text{ such that } \neg Q(x) $$ + +--- + +Page 146 + +**Theorem 3.2.2 Negation of an Existential Statement** + +The negation of a statement of the form + +$$ \exists \text{ in } D \text{ such that } Q(x) $$ + +is logically equivalent to a statement of the form + +$$ \forall x \text{ in } D, \neg Q(x) $$ + +Symbolically, + +$$ \neg(\exists x \in D \text{ such that } Q(x)) \equiv \forall x \in D, \neg Q(x) $$ + +--- + +Page 148 + +**Negation of a Universal Conditional Statement** + +$$ \neg(\forall x, \text{ if } P(x) \text{ then } Q(x)) \equiv \exists x \text{ such that } P(x) \text{ and } \neg Q(x) $$ + +$$ \neg(\forall x, P(x) \to Q(x)) \equiv \exists x, (P(x) \wedge \neg Q(x)) $$ + +--- + +Page 150 + +**Definition** + +Consider a statement of the form +$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$. + +1. Its **contrapositive** is the statement + $\forall x \in D, \text{ if } \neg Q(x) \text{ then } \neg P(x)$. + +2. Its **converse** is the statement + $\forall x \in D, \text{ if } Q(x) \text{ then } P(x)$. + +3. Its **inverse** is the statement + $\forall x \in D, \text{ if } \neg P(x) \text{ then } \neg Q(x)$. + +--- + +Page 151 + +**Definition** + +- "$\forall x, r(x)$ is a **sufficient condition** for $s(x)$" means + "$\forall x, \text{ if } r(x) \text{ then } s(x)$." + +- "$\forall x, r(x)$ is a **necessary condition** for $s(x)$" means + "$\forall x, \text{ if } \neg r(x) \text{ then } \neg s(x)$" or, equivalently, + "$\forall x, \text{ if } s(x) \text{ then } r(x)$." + +- "$\forall x, r(x)$ **only if** $s(x)$" means + "$\forall x, \text{ if } \neg s(x) \text{ then } \neg r(x)$" or, equivalently, + "$\forall x, \text{ if } r(x) \text{ then } s(x)$." diff --git a/chapter_3/test_yourself.md b/chapter_3/test_yourself.md index 36c9719..df3f045 100644 --- a/chapter_3/test_yourself.md +++ b/chapter_3/test_yourself.md @@ -25,3 +25,41 @@ true; every $x$ in $D$. only if, $Q(x)$ is _______ for _______. true; at least one $x$ in $D$. + +--- + +**Test Yourself** + +Page 152 + +1. A negation for "All $R$ have property $S$" is "There is _______ $R$ that + _______." + +exists at least one; does not have property $S$. + +2. A negation for "Some $R$ have property $S$" is "_______." + +"No $R$ have property $S$." + +3. A negation for "For every $x$, if $x$ has property $P$ then $x$ has property + $Q$" is "_______." + +"There exists at least one $x$ such that $x$ has property $P$ and $x$ does not +have property $Q$." + +4. The converse of "For every $x$, if $x$ has property $P$ then $x$ has property + $Q$" is "_______." + +"For every $x$, if $x$ has property $Q$ then $x$ has property $P$." + +5. The contrapositive of "For every $x$, if $x$ has property $P$ then $x$ has + property $Q$" is "_______." + +"For every $x$, if $x$ does not have property $Q$, then $x$ does not have +property $P$." + +6. The inverse of "For every $x$, if $x$ has property $P$ then $x$ has property + $Q$" is "_______." + +"For every $x$, if $x$ does not have property $P$, then $x$ does not have +property $Q$."