🚧 Setup for 3.3

chapter_3/exercises.md

@@ -1547,3 +1547,392 @@ This is:
 $$ F \to S $$
 
 This is the converse statement, and is not logically equivalent to the original.
+
+---
+
+**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. $P(\text{Tokyo}, \text{Japan})$
+
+b. $P(\text{Athens}, \text{Egypt})$
+
+c. $P(\text{Paris}, \text{France})$
+
+d. $P(\text{Miami}, \text{Brazil})$
+
+2. Let $G(x, y)$ be "$x^2 > y$." Indicate which of the following statements are
+   true and which are false.
+
+a. $G(2, 3)$
+
+b. $G(1, 1)$
+
+c. $G(\dfrac{1}{2}, \dfrac{1}{2})$
+
+d. $G(-2, 2)$
+
+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. $x = 2$
+
+b. $x = -1$
+
+c. $x = \dfrac{3}{4}$
+
+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. $x = 15.83$
+
+b. $x = 10^8$
+
+c. $x = 10^{10^{10}}$
+
+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.
+
+6. For every square $x$ there is a circle $y$ such that $x$ and $y$ have
+   different colors and $y$ is above $x$.
+
+7. There is a triangle $x$ such that for every square $y$, $x$ is above $y$.
+
+8. There is a triangle $x$ such that for every circle $y$, $y$ is above $x$.
+
+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 $x + y = 0$.
+
+b. $\exists x$ in $D$ such that $\forall y$ in $E$, $x + y = y$.
+
+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 $D$.
+
+b. $\forall$ student $S$, $\exists$ a salad $T$ such that $S$ chose $T$.
+
+c. $\exists$ a dessert $D$ such that $\forall$ student $S$, $S$ chose $D$.
+
+d. $\exists$ a beverage $B$ such that $\forall$ student $D$, $D$ chose $B$.
+
+e. $\exists$ an item $I$ such that $\forall$ student $S$, $S$ did not choose
+$I$.
+
+f. $\exists$ a station $Z$ such that $\forall$ student $S$, $\exists$ an item
+$I$ such that $S$ chose $I$ from $Z$.
+
+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 $V(s, \text{Casablanca})$.
+
+b. $\forall s \in S, V(s, \text{Star Wars})$.
+
+c. $\forall s \in S, \exists m \in M \text{ such that } V(s, m)$.
+
+d. $\exists m \in M \text{ such that } \forall s \in S, V(s, m)$.
+
+e.
+$\exists s \in S, \exists t \in S, \text{ and } \exists m \in M \text{ such that } s \neq t \text{ and } V(s, m) \wedge V(t, m)$.
+
+f.
+$\exists s \in S \text{ and } \exists t \in S \text{ such that } s \neq t \text{ and } \forall m \in M, V(s, m) \to V(t, m)$.
+
+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 $x + y = 1$.
+
+b. $\exists x$ in $D$ such that $\forall y$ in $E$, $x + y = -y$.
+
+c. $\forall x$ in $D$, $\exists y$ in $E$ such that $xy \geq y$.
+
+d. $\exists x$ in $D$ such that $\forall y$ in $E$, $x \leq y$.
+
+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 $C$.
+
+14. $\exists$ a book $b$ such that $\forall$ person $p$, $p$ has read $b$.
+
+15. $\forall$ odd integer $n$, $\exists$ an integer $k$ such that $n = 2k + 1$.
+
+16. $\exists$ a real number $u$ such that $\forall$ real number $v$, $uv = v$.
+
+17. $\forall r \in \mathbb{Q}$, $\exists$ integers $a$ and $b$ such that
+    $r = \dfrac{a}{b}$.
+
+18. $\forall x \in \mathbb{R}$, $\exists$ a real number $y$ such that
+    $x + y = 0$.
+
+19. $\exists x \in \mathbb{R}$ such that for every real number $y$, $x + y = 0$.
+
+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.
+
+(2) There is a triangle $x$ such that for every square $y$, $x$, and $y$ have
+different colors.
+
+b.
+
+(1) For every circle $y$ there is a square $x$ such that $x$ and $y$ have the
+same color.
+
+(2) There is a square $x$ such that for every circle $y$, $x$ and $y$ have the
+same color.
+
+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. $2x + y = 7$
+
+b. $y + x = x + y$
+
+c. $x^2 - 2xy + y^2 = 0$
+
+d. $(x - 5)(y - 1) = 0$
+
+e. $x^2 + y^2 = -1$
+
+In 22 and 23, rewrite each statement without using variables or the symbol
+$\forall$ or $\exists$. Indicate whether the statement is true or false.
+
+22.
+
+a. $\forall$ real number $x$, $\exists$ a real number $y$ such that $x + y = 0$.
+
+b. $\exists$ a real number $y$ such that $\forall$ real number $x$, $x + y = 0$.
+
+23.
+
+a. $\forall$ nonzero real number $r$, $\exists$ a real number $s$ such that
+$rs = 1$.
+
+b. $\exists$ a real number $r$ such that $\forall$ nonzero real number $s$,
+$rs = 1$.
+
+24. Use the laws for negating universal and existential statements to derive the
+    following rules:
+
+a.
+$\neg(\forall x \in D(\forall y \in E(P(x, y)))) \equiv \exists x \in D(\exists y \in E(\neg P(x, y)))$
+
+b.
+$\neg(\exists x \in D(\exists y \in E(P(x, y)))) \equiv \forall x \in D(\forall y \in E(\neg P(x, y)))$
+
+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 $y$.
+
+26. $\forall$ circle $x$ and $\forall$ triangle $y$, $x$ is 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.
+
+28. $\exists$ a triangle $x$ and $\exists$ a square $y$ such that $x$ is above
+    $y$ and $x$ and $y$ have 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 $x < y$.
+
+30. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}^{-}$ (the set
+    of negative real numbers), $x > y$.
+
+31. Consider the statement "Everybody is older than somebody." Rewrite this
+    statement in the form "$\forall$ people $x$, $\exists$ ______."
+
+32. Consider the statement "Somebody is older than everybody." Rewrite this
+    statement in the form "$\exists$ a person $x$ such that $\forall$ ______."
+
+In 33-39, (a) rewrite the statement formally using quantifiers and variables,
+and (b) write a negation for the statement.
+
+33. Everybody loves somebody.
+
+34. Somebody loves everybody.
+
+35. Everybody trusts somebody.
+
+36. Somebody trusts everybody.
+
+37. Any even integer equals twice some integer.
+
+38. Every action has an equal and opposite reaction.
+
+39. There is a program that gives the correct answer to every question that is
+    posed to it.
+
+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.
+
+b. There is a time for every purpose under heaven.
+
+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
+$x = y + 1$.
+
+b. $\forall x \in \mathbb{Z}, \exists y \in \mathbb{Z}$ such that $x = y + 1$.
+
+c. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}, x = y + 1$ .
+
+d. $\forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+}$ such that
+$xy = 1$.
+
+e. $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}$ such that $xy = 1$.
+
+f. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}, x + y = y$.
+
+g. $\forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+}$ such that
+$y < x$.
+
+h. $\exists x \in \mathbb{R}^{+}$ such that
+$\forall y \in \mathbb{R}^{+}, x \leq y$.
+
+42. Write the negation of the definition of limit of a sequence given in Example
+    3.3.7.
+
+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.
+
+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 $y$, $xy = y$.
+
+b. $\exists !$ integer $x$ such that $\dfrac{1}{x}$ is an integer.
+
+c. $\forall$ real number $x$, $\exists !$ real number $y$ such that $x + y = 0$.
+
+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 $\exists !$.)
+
+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 $y$.
+
+47. There is a triangle $x$ such that for every circle $y$, $x$ is above $y$.
+
+48. For every circle $x$, there is a square $y$ such that $y$ is 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 $x$.
+
+50. For every object $x$, if $x$ is a triangle then there is a square $y$ such
+    that $y$ 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 $x$.
+
+52. For every circle $x$ and for every triangle $y$, $x$ is to the right of $y$.
+
+53. There is a circle $x$ and there is a square $y$ such that $x$ 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 $y$.
+
+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. $\forall x \in D, (P(x) \wedge Q(x)) \text{ and } (\forall x \in D, P(x)) \wedge (\forall x \in D, Q(x))$
+
+56. $\exists x \in D, (P(x) \wedge Q(x)) \text{ and } (\exists x \in D, P(x)) \wedge (\exists x \in D, Q(x))$
+
+57. $\forall x \in D, (P(x) \vee Q(x)) \text{ and } (\forall x \in D, P(x)) \vee (\forall x \in D, Q(x))$
+
+58. $\exists x \in D, (P(x) \vee Q(x)) \text{ and } (\exists x \in D, P(x)) \vee (\exists x \in D, Q(x))$
+
+In 59-61, find the answers Prolog would give if the following questions were
+added to the program given in Example 3.3.11.
+
+59.
+
+a. $?\text{isabove}(b_1, w_1)$
+
+b. $?\text{color}(X, white)$
+
+c. $?\text{isabove}(X, b_3)$
+
+60.
+
+a. $?\text{isabove}(w_1, g)$
+
+b. $?\text{color}(w_2, blue)$
+
+c. $?\text{isabove}(X, b_1)$
+
+61.
+
+a. $?\text{isabove}(w_2, b_3)$
+
+b. $?\text{color}(X, gray)$
+
+c. $?\text{isabove}(g, X)$

chapter_3/test_yourself.md

@@ -63,3 +63,41 @@ property $P$."
 
 "For every $x$, if $x$ does not have property $P$, then $x$ does not have
 property $Q$."
+
+---
+
+**Test Yourself**
+
+Page 165
+
+1. To establish the truth of a statement of the form
+   "$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x)$,"
+   you imagine that someone has given you an element $x$ from $D$ but that you
+   have no control over what that element is. Then you need to find _______ with
+   the property that the $x$ the person gave you together with the _______ you
+   subsequently found satisfy _______.
+
+2. To establish the truth of a statement of the form
+   "$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$,"
+   you need to find _______ so that no matter what _______ a person might
+   subsequently give you, _______ will be true.
+
+3. Consider the statement
+   "$\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
+   A negation for this statement is "_______."
+
+4. Consider the statement
+   "$\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$."
+   A negation for this statement is "_______."
+
+5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the
+   statement
+   "$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$"
+   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.