From 732483ab1cfde3d2122edc2c2734c057e14afec0 Mon Sep 17 00:00:00 2001 From: tomit4 Date: Fri, 5 Jun 2026 23:42:01 -0700 Subject: [PATCH] :construction: Fin 3.3 --- chapter_3/exercises.md | 683 ++++++++++++++++++++++++++++++++++++- chapter_3/test_yourself.md | 11 + 2 files changed, 691 insertions(+), 3 deletions(-) diff --git a/chapter_3/exercises.md b/chapter_3/exercises.md index 56cae21..2207be1 100644 --- a/chapter_3/exercises.md +++ b/chapter_3/exercises.md @@ -1560,80 +1560,190 @@ Page 166 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 @@ -1641,49 +1751,137 @@ $I$ such that $S$ chose $I$ from $Z$. 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 @@ -1697,17 +1895,46 @@ 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: @@ -1721,14 +1948,102 @@ 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. @@ -1736,25 +2051,48 @@ $\forall$ 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. + 23. 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 @@ -1762,47 +2100,131 @@ 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 @@ -1812,34 +2234,83 @@ and (b) write a negation for the statement. 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 $\delta > 0$ @@ -1851,6 +2322,21 @@ $$ L - \varepsilon < f(x) < L + \varepsilon $$ 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 @@ -1858,14 +2344,33 @@ write the negation of the definition. 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 @@ -1875,28 +2380,159 @@ 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 @@ -1904,12 +2540,35 @@ 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. @@ -1917,22 +2576,40 @@ added to the program given in Example 3.3.11. a. $?\text{isabove}(b_1, w_1)$ -b. $?\text{color}(X, white)$ +True. + +b. $?\text{color}(X, \text{white})$ + +$X = w_2, X = w_1$ c. $?\text{isabove}(X, b_3)$ +$X = b_2, X = w_2$ + 60. a. $?\text{isabove}(w_1, g)$ -b. $?\text{color}(w_2, blue)$ +False. + +b. $?\text{color}(w_2, \text{blue})$ + +False. c. $?\text{isabove}(X, b_1)$ +$X = g$ + 61. a. $?\text{isabove}(w_2, b_3)$ -b. $?\text{color}(X, gray)$ +True. + +b. $?\text{color}(X, \text{gray})$ + +$X = g$ c. $?\text{isabove}(g, X)$ + +$X = b_1, X = w_1$ diff --git a/chapter_3/test_yourself.md b/chapter_3/test_yourself.md index 77de8ea..05b9236 100644 --- a/chapter_3/test_yourself.md +++ b/chapter_3/test_yourself.md @@ -77,19 +77,27 @@ Page 165 the property that the $x$ the person gave you together with the _______ you subsequently found satisfy _______. +$y \in E$; $y$; $P(x, y)$ + 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. +$x \in D$; $y \in E$; $P(x, y)$ + 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 "_______." +"$\exists x$ such that $\forall y$, the property $P(x, y)$ is false." + 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 "_______." +"$\forall x, \exists y$ such that the property $P(x, y)$ is false." + 5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the statement "$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$" @@ -101,3 +109,6 @@ a. is true. b. is false. c. may be true or may be false. + +c is the answer, it may be true or false depending on the nature of the property +involving $x$, and $y$. In other words, it relies on what $P(x, y)$ states.