🚧 Setup for 3.3
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@ -1547,3 +1547,392 @@ This is:
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$$ F \to S $$
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This is the converse statement, and is not logically equivalent to the original.
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---
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**Exercise Set 3.3**
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Page 166
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1. Let $C$ be the set of cities in the world, let $N$ be the set of nations in
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the world, and let $P(c, n)$ be "$c$ is the capital city of $n$." Determine
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the truth values of the following statements.
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a. $P(\text{Tokyo}, \text{Japan})$
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b. $P(\text{Athens}, \text{Egypt})$
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c. $P(\text{Paris}, \text{France})$
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d. $P(\text{Miami}, \text{Brazil})$
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2. Let $G(x, y)$ be "$x^2 > y$." Indicate which of the following statements are
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true and which are false.
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a. $G(2, 3)$
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b. $G(1, 1)$
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c. $G(\dfrac{1}{2}, \dfrac{1}{2})$
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d. $G(-2, 2)$
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3. The following statement is true: "$\forall$ nonzero number $x$, $\exists$ a
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real number $y$ such that $xy = 1$." For each $x$ given below, find a $y$ to
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make the predicate "$xy = 1$" true.
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a. $x = 2$
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b. $x = -1$
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c. $x = \dfrac{3}{4}$
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4. The following statement is true: "$\forall$ real number $x$, $\exists$ an
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integer $n$ such that $n > x$.". For each $x$ given below, find an $n$ to
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make the predicate $n > x$ true.
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a. $x = 15.83$
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b. $x = 10^8$
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c. $x = 10^{10^{10}}$
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The statements in exercises 5-8 refer to the Tarski world given in Figure 3.3.1.
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Explain why each is true.
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5. For every circle $x$ there is a square $y$ such that $x$ and $y$ have the
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same color.
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6. For every square $x$ there is a circle $y$ such that $x$ and $y$ have
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different colors and $y$ is above $x$.
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7. There is a triangle $x$ such that for every square $y$, $x$ is above $y$.
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8. There is a triangle $x$ such that for every circle $y$, $y$ is above $x$.
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9. Let $D = E = \{-2, -1, 0, 1, 2\}$. Explain why the following statements are
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true.
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a. $\forall x$ in $D$, $\exists y$ such that $x + y = 0$.
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b. $\exists x$ in $D$ such that $\forall y$ in $E$, $x + y = y$.
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10. This exercise refers to Example 3.3.3. Determine whether each of the
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following statements is true or false.
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a. $\forall$ student $S$, $\exists$ a dessert $D$ such that $S$ chose $D$.
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b. $\forall$ student $S$, $\exists$ a salad $T$ such that $S$ chose $T$.
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c. $\exists$ a dessert $D$ such that $\forall$ student $S$, $S$ chose $D$.
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d. $\exists$ a beverage $B$ such that $\forall$ student $D$, $D$ chose $B$.
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e. $\exists$ an item $I$ such that $\forall$ student $S$, $S$ did not choose
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$I$.
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f. $\exists$ a station $Z$ such that $\forall$ student $S$, $\exists$ an item
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$I$ such that $S$ chose $I$ from $Z$.
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11. Let $S$ be the set of students at your school, let $M$ be the set of movies
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that have ever been released, and let $V(s, m)$ be "student $s$ has seen
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movie $m$." Rewrite each of the following statements without using the
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symbol $\forall$, the symbol $\exists$, or variables.
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a. $\exists s \in S$ such that $V(s, \text{Casablanca})$.
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b. $\forall s \in S, V(s, \text{Star Wars})$.
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c. $\forall s \in S, \exists m \in M \text{ such that } V(s, m)$.
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d. $\exists m \in M \text{ such that } \forall s \in S, V(s, m)$.
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e.
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$\exists s \in S, \exists t \in S, \text{ and } \exists m \in M \text{ such that } s \neq t \text{ and } V(s, m) \wedge V(t, m)$.
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f.
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$\exists s \in S \text{ and } \exists t \in S \text{ such that } s \neq t \text{ and } \forall m \in M, V(s, m) \to V(t, m)$.
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12. Let $D = E = \{-2, -1, 0, 1, 2\}$. Write negations for each of the following
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statements and determine which is true, the given statement or its negation.
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a. $\forall x$ in $D$, $\exists y$ such that $x + y = 1$.
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b. $\exists x$ in $D$ such that $\forall y$ in $E$, $x + y = -y$.
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c. $\forall x$ in $D$, $\exists y$ in $E$ such that $xy \geq y$.
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d. $\exists x$ in $D$ such that $\forall y$ in $E$, $x \leq y$.
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In each of 13-19, (a) rewrite the statement in English without using the symbol
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$\forall$ or $\exists$ or variables and expressing your answer as simply as
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possible, and (b) write a negation for the statement.
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13. $\forall$ color $C$, $\exists$ an animal $A$ such that $A$ is colored $C$.
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14. $\exists$ a book $b$ such that $\forall$ person $p$, $p$ has read $b$.
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15. $\forall$ odd integer $n$, $\exists$ an integer $k$ such that $n = 2k + 1$.
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16. $\exists$ a real number $u$ such that $\forall$ real number $v$, $uv = v$.
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17. $\forall r \in \mathbb{Q}$, $\exists$ integers $a$ and $b$ such that
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$r = \dfrac{a}{b}$.
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18. $\forall x \in \mathbb{R}$, $\exists$ a real number $y$ such that
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$x + y = 0$.
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19. $\exists x \in \mathbb{R}$ such that for every real number $y$, $x + y = 0$.
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20. Recall that reversing the order of the quantifiers in a statement with two
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different quantifiers may change the truth value of the statement - but it
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does not necessarily do so. All the statements in the pairs below refer to
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the Tarski world of Figure 3.3.1. In each pair, the order of the quantifiers
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is reversed but everything else is the same. For each pair, determine
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whether the statements have the same or opposite truth values. Justify your
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answers.
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a.
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(1) For every square $y$ there is a triangle $x$ such that $x$ and $y$ have
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different colors.
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(2) There is a triangle $x$ such that for every square $y$, $x$, and $y$ have
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different colors.
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b.
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(1) For every circle $y$ there is a square $x$ such that $x$ and $y$ have the
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same color.
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(2) There is a square $x$ such that for every circle $y$, $x$ and $y$ have the
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same color.
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21. For each of the following equations, determine which of the following
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statements are true:
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(1) For every real number $x$, there exists a real number $y$ such that the
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equation is true.
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(2) There exists a real number $x$, such that for every real number $y$, the
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equation is true.
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Note that it is possible for both statements to be true or for both to be false.
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a. $2x + y = 7$
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b. $y + x = x + y$
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c. $x^2 - 2xy + y^2 = 0$
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d. $(x - 5)(y - 1) = 0$
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e. $x^2 + y^2 = -1$
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In 22 and 23, rewrite each statement without using variables or the symbol
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$\forall$ or $\exists$. Indicate whether the statement is true or false.
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22.
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a. $\forall$ real number $x$, $\exists$ a real number $y$ such that $x + y = 0$.
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b. $\exists$ a real number $y$ such that $\forall$ real number $x$, $x + y = 0$.
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23.
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a. $\forall$ nonzero real number $r$, $\exists$ a real number $s$ such that
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$rs = 1$.
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b. $\exists$ a real number $r$ such that $\forall$ nonzero real number $s$,
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$rs = 1$.
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24. Use the laws for negating universal and existential statements to derive the
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following rules:
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a.
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$\neg(\forall x \in D(\forall y \in E(P(x, y)))) \equiv \exists x \in D(\exists y \in E(\neg P(x, y)))$
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b.
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$\neg(\exists x \in D(\exists y \in E(P(x, y)))) \equiv \forall x \in D(\forall y \in E(\neg P(x, y)))$
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Each statement in 25-28 refers to the Tarski world of Figure 3.3.1. For each,
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(a) determine whether the statement is true or false and justify your answer,
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and (b) write a negation for the statement (referring, if you wish, to the
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result in exercise 24).
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25. $\forall$ circle $x$ and $\forall$ square $y$, $x$ is above $y$.
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26. $\forall$ circle $x$ and $\forall$ triangle $y$, $x$ is above $y$.
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27. $\exists$ a circle $x$ and $\exists$ a square $y$ such that $x$ is above $y$
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and $x$ and $y$ have different colors.
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28. $\exists$ a triangle $x$ and $\exists$ a square $y$ such that $x$ is above
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$y$ and $x$ and $y$ have the same color.
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For each of the statements in 29 and 30, (a) write a new statement by
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interchanging the symbols $\forall$ and $\exists$, and (b) state which is true:
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the given statement, the version with interchanged quantifiers, neither or both.
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29. $\forall x \in \mathbb{R}, \exists y \in \mathhbb{R}$ such that $x < y$.
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30. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}^{-}$ (the set
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of negative real numbers), $x > y$.
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31. Consider the statement "Everybody is older than somebody." Rewrite this
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statement in the form "$\forall$ people $x$, $\exists$ ______."
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32. Consider the statement "Somebody is older than everybody." Rewrite this
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statement in the form "$\exists$ a person $x$ such that $\forall$ ______."
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In 33-39, (a) rewrite the statement formally using quantifiers and variables,
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and (b) write a negation for the statement.
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33. Everybody loves somebody.
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34. Somebody loves everybody.
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35. Everybody trusts somebody.
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36. Somebody trusts everybody.
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37. Any even integer equals twice some integer.
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38. Every action has an equal and opposite reaction.
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39. There is a program that gives the correct answer to every question that is
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posed to it.
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40. In informal speech most sentences of the form "There is ______ every ______"
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are intended to be understood as meaning "$\forall$ ______ $\exists$
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______," even though the existential quantifier _there is_ comes before the
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universal quantifier _every_. Note that this interpretation applies to the
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following well-known sentences. Rewrite them using quantifiers and
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variables.
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a. There is a sucker born every minute.
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b. There is a time for every purpose under heaven.
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41. Indicate which of the following statements are true and which are false.
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Justify your answers as best you can.
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a. $\forall x \in \mathbb{Z}^{+}, \exists y \in \mathbb{Z}^{+}$ such that
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$x = y + 1$.
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b. $\forall x \in \mathbb{Z}, \exists y \in \mathbb{Z}$ such that $x = y + 1$.
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c. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}, x = y + 1$ .
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d. $\forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+}$ such that
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$xy = 1$.
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e. $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}$ such that $xy = 1$.
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f. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}, x + y = y$.
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g. $\forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+}$ such that
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$y < x$.
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h. $\exists x \in \mathbb{R}^{+}$ such that
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$\forall y \in \mathbb{R}^{+}, x \leq y$.
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42. Write the negation of the definition of limit of a sequence given in Example
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3.3.7.
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43. The following is the definition for $\lim\limits_{x \to a}f(x) = L$:
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For every real number $\varepsilon > 0$, there exists a real number $\delta > 0$
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such that for every real number $x$, if $a - \delta < x < a + \delta$ and
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$x \neq a$ then
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$$ L - \varepsilon < f(x) < L + \varepsilon $$
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Write what it means for $\lim\limits_{x \to a}f(x) \neq L$. In other words,
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write the negation of the definition.
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44. The notation $\exists !$ stands for the words "there exists a unique." Thus,
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for instance, "$\exists ! x$ such that $x$ is prime and $x$ is even" means
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that there is one and only one even prime number. Which of the following
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statements are true and which are false?
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a. $\exists !$ real number $x$ such that $\forall$ real number $y$, $xy = y$.
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b. $\exists !$ integer $x$ such that $\dfrac{1}{x}$ is an integer.
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c. $\forall$ real number $x$, $\exists !$ real number $y$ such that $x + y = 0$.
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45. Suppose that $P(x)$ is a predicate and $D$ is the domain of $x$. Rewrite the
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statement "$\exists ! x \in D \text{ such that } P(x)$" without using the
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symbol $\exists !$. (See exercise 44 for the meaning of $\exists !$.)
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In 46-54, refer to the Tarski world given in Figure 3.1.1, which is shown again
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here for reference. The domains of all variables consist of all the objects in
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the Tarski world. For each statement, (a) indicate whether the statement is true
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or false and justify your answer, (b) write the given statement using the formal
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logical notation illustrated in Example 3.3.10, and (c) write a negation for the
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given statement using the formal logical notation of Example 3.3.10.
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46. There is a triangle $x$ such that for every square $y$, $x$ is above $y$.
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47. There is a triangle $x$ such that for every circle $y$, $x$ is above $y$.
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48. For every circle $x$, there is a square $y$ such that $y$ is to the right of
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$x$.
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49. For every object $x$, if $x$ is a circle then there is a square $y$ such
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that $y$ has the same color as $x$.
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50. For every object $x$, if $x$ is a triangle then there is a square $y$ such
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that $y$ is below $x$.
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51. There is a square $x$ such that for every triangle $y$, if $y$ is above $x$
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then $y$ has the same color as $x$.
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52. For every circle $x$ and for every triangle $y$, $x$ is to the right of $y$.
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53. There is a circle $x$ and there is a square $y$ such that $x$ and $y$ have
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the same color.
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54. There is a circle $x$ and there is a triangle $y$ such that $x$ has the same
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color as $y$.
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Let $P(x)$ and $Q(x)$ be predicates and suppose $D$ is the domain of $x$. In
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55-58, for the statement forms in each pair, determine whether (a) they have the
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same truth value for every choice of $P(x)$, $Q(x)$ and $D$, or (b) there is a
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choice of $P(x)$, $Q(x)$, and $D$ for which they have opposite truth values.
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55. $\forall x \in D, (P(x) \wedge Q(x)) \text{ and } (\forall x \in D, P(x)) \wedge (\forall x \in D, Q(x))$
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56. $\exists x \in D, (P(x) \wedge Q(x)) \text{ and } (\exists x \in D, P(x)) \wedge (\exists x \in D, Q(x))$
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57. $\forall x \in D, (P(x) \vee Q(x)) \text{ and } (\forall x \in D, P(x)) \vee (\forall x \in D, Q(x))$
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58. $\exists x \in D, (P(x) \vee Q(x)) \text{ and } (\exists x \in D, P(x)) \vee (\exists x \in D, Q(x))$
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In 59-61, find the answers Prolog would give if the following questions were
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added to the program given in Example 3.3.11.
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59.
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a. $?\text{isabove}(b_1, w_1)$
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b. $?\text{color}(X, white)$
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c. $?\text{isabove}(X, b_3)$
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60.
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a. $?\text{isabove}(w_1, g)$
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b. $?\text{color}(w_2, blue)$
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c. $?\text{isabove}(X, b_1)$
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61.
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a. $?\text{isabove}(w_2, b_3)$
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b. $?\text{color}(X, gray)$
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c. $?\text{isabove}(g, X)$
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@ -63,3 +63,41 @@ property $P$."
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"For every $x$, if $x$ does not have property $P$, then $x$ does not have
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property $Q$."
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---
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**Test Yourself**
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Page 165
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1. To establish the truth of a statement of the form
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"$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x)$,"
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you imagine that someone has given you an element $x$ from $D$ but that you
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have no control over what that element is. Then you need to find _______ with
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the property that the $x$ the person gave you together with the _______ you
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subsequently found satisfy _______.
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2. To establish the truth of a statement of the form
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"$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$,"
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you need to find _______ so that no matter what _______ a person might
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subsequently give you, _______ will be true.
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3. Consider the statement
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"$\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
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A negation for this statement is "_______."
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4. Consider the statement
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"$\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
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A negation for this statement is "_______."
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5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the
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statement
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"$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$"
|
||||
is true. Then the statement
|
||||
"$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$"
|
||||
|
||||
a. is true.
|
||||
|
||||
b. is false.
|
||||
|
||||
c. may be true or may be false.
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue