From e7ef70ff03515f50d24426645ec96101ec60ca36 Mon Sep 17 00:00:00 2001 From: tomit4 Date: Tue, 18 Aug 2026 23:47:54 -0700 Subject: [PATCH] :construction: Setup for 8.3 --- chapter_8/exercises.md | 402 +++++++++++++++++++++++++++++++++++++ chapter_8/notes.md | 343 +++++++++++++++++++++++++++++++ chapter_8/test_yourself.md | 24 +++ 3 files changed, 769 insertions(+) diff --git a/chapter_8/exercises.md b/chapter_8/exercises.md index 782006d..5383db8 100644 --- a/chapter_8/exercises.md +++ b/chapter_8/exercises.md @@ -1957,3 +1957,405 @@ Omitted. $$ A = \{a[1], a[2], \dots, a[n]\} $$ Omitted. + +--- + +Page 543 + +**Exercise Set 8.3** + +1. Suppose that $S = \{a, b, c, d, e\}$ and $R$ is a relation on $S$ such that + $a R b$, $b R c$, and $d R e$. List all of the following that must be true if + $R$ is (a) reflexive (but not symmetric or transitive), (b) symmetric (but + not reflexive or ransitive), \(c\) transitive (but not reflexive or + symmetric), and (d) an equivalence relation. + +$$ c R b \quad c R c \quad a R c \quad b R a $$ + +$$ a R d \quad e R a \quad e R d \quad c R a $$ + +2. Each of the following partitions of $\{0, 1, 2, 3, 4\}$ induces a relation + $R$ on $\{0, 1, 2, 3, 4\}$. In each case, find the ordered pairs in $R$. + +a. $\{0, 2\}, \{1\}, \{3, 4\}$ + +b. $\{0\}, \{1, 3, 4\}, \{2\}$ + +c. $\{0\}$, $\{1, 2, 3, 4\}$ + +In each of 3-6, the relation $R$ is an equivalence relation on $A$. As in +example 8.3.5, first find the specified equivalence classes. Then state the +number of distinct equivalence classes for $R$ and list them. + +3. + +$$ A = \{0, 1, 2, 3, 4\} $$ + +$$ R = \{(0, 0), (0, 4), (1, 1), (1, 3), (2, 2), (3, 1), (3, 3), (4, 0), (4, 4)\} $$ + +equivalence classes: $[0], [1], [2], [3]$ + +4. + +$$ A = \{a, b, c, d\} $$ + +$$ R = \{(a, a), (b, b), (b, d), (c, c), (d, b), (d, d)\} $$ + +equivalence classes: $[a], [b], [c], [d]$ + +5. + +$$ A = \{1, 2, 3, 4, \dots, 20\} $$ + +$R$ is defined on $A$ as follows: + +$$ \text{For all } x, y \in A, x R y \Leftrightarrow 4 | (x - y) $$ + +equivalence classes: $[1], [2], [3], [4], [5]$ + +6. + +$$ A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} $$ + +$R$ is defined on $A$ as follows: + +$$ \text{For all } x, y \in A, x R y \Leftrightarrow 3 | (x - y) $$ + +equivalence classes: $[0], [1], [2], [3]$ + +In each of 7-14, the relation $R$ is an equivalence relation on the set $A$. +Find the distinct equivalence classes of $R$. + +7. $A = \{(1, 3), (2, 4), (-4, -8), (3, 9), (1, 5), (3, 6)\}$. $R$ is defined on + $A$ as follows: For every $(a, b), (c, d) \in A$, + +$$ (a, b) R (c, d) \Leftrightarrow ad = bc $$ + +8. $X = \{a, b, c\}$ and $A = \mathscr{P}(X)$. $R$ is defined on $A$ as follows: + For all sets $u$ and $v$ in $\mathscr{P}(X)$, + +$$ u R v \Leftrightarrow N(u) = N(v) $$ + +(That is, the number of elements in $u$ equals the number of elements in $v$.) + +9. $X = \{-1, 0, 1\}$ and $A = \mathscr{P}(X)$. $R$ is defined on + $\mathscr{P}(X)$ as follows: For all sets $s$ and $t$ in $\mathscr{P}(X)$, + +$$ s R t \Leftrightarrow \text{ the sum of the elements in } s \text{ equals the sum of the elements in } t $$ + +10. $A = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$. $R$ is defined on $A$ as + follows: For all $m, n \in \mathbb{Z}$, + +$$ m R n \Leftrightarrow 3 |(m^2 - n^2) $$ + +11. $A = \{-4, -3, -2< -1, 0, 1, 2, 3, 4\}$. $R$ is defined on $A$ as follows: + For every $(m, n) \in A$, + +$$ m R n \Leftrightarrow 4 | (m^2 - n^2) $$ + +12. $A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$. $R$ is defined on $A$ as follows: + For all $(m, n) \in A$, + +$$ m R n \Leftrightarrow 5 | (m^2 - n^2) $$ + +13. $A$ is the set of all strings of length 4 in _a_'s and _b_'s. $R$ is defined + on $A$ as follows: For all strings $s$ and $t$ in $A$, + +$$ s R t \Leftrightarrow s \text{ has the same first two characters as } t $$ + +14. $A$ is the set of all strings of 0's, 1's, and 2's that have length 4 and + for which the sum of the characters in the string is less than or equal + to 2. $R$ is defined on $A$ as follows: For every $s, t \in A$, + +$$ s R t \Leftrightarrow \text{ the sum of the characters of } s \text{ equals the sum of the characters of } t $$ + +15. Determine which of the following congruence relations are true and which are + false. + +a. $17 \equiv 2 (\mod 5)$ + +b. $4 \equiv -5 (\mod 7)$ + +c. $-2 \equiv -8 (\mod 3)$ + +d. $-6 \equiv 22 (\mod 2)$ + +16. + +a. Let $R$ be the relation of congruence modulo 3. Which of the following +equivalence classes are equal? + +$$ [7], [-4], [-6], [17], [4], [27], [19] $$ + +b. Let $R$ be the relation of congruence modulo 7. Which of the following +equivalence classes are equal? + +$$ [35], [3], [-7], [12], [0], [-2], [17] $$ + +17. + +a. Prove that for all integers $m$ and $n$, $m \equiv n (\mod 3)$ if, and only +if, $m \mod 3 = n \mod 3$. + +b. Prove that for all integers $m$ and $n$ and any positive integer $d$, +$m \equiv n (\mod d)$ if, and only if, $m \mod d = n \mod d$. + +18. + +a. Give an example of two sets that are distinct but not disjoint. + +b. Find sets $A_1$ and $A_2$ and elements $x$, $y$, and $z$ such that $x$ and +$y$ are in $A_1$ and $y$ and $z$ are in $A_2$ but $x$ and $z$ are not both in +either of the sets $A_1$ or $A_2$. + +In 19-31, (1) prove that the relation is an equivalence relation, and (2) +describe the distinct equivalence classes of each relation. + +19. $A$ is the set of all students at your college. + +a. $R$ is the relation defined on $A$ a follows: For every $x$ and $y$ in $A$, + +$$ x R y \Leftrightarrow x \text{ has the same major (or double major) as } y $$ + +(Assume "undeclared" is a major.) + +b. $S$ is the relation defined on $A$ as follows: For every $x, y \in A$, + +$$ x S y \Leftrightarrow x \text{ is the same age as } y $$ + +20. $E$ is the relation defined on $\mathbb{Z}$ as follows: + +$$ \text{For every } m, n \in \mathbb{Z}, m E n \Leftrightarrow 4 | (m - n) $$ + +21. $R$ is the relation defined on $\mathbb{Z}$ as follows: + +$$ \text{For every } m, n \in \mathbb{Z}, m R n \Leftrightarrow 7m - 5n \text{ is even} $$ + +22. Let $A$ be the set of all statement forms in three variables $p$, $q$, and + $r$. $\mathbf{R}$ is the relation defined on $A$ as follows: For all $P$ and + $Q$ in $A$, + +$$ P \mathbf{R} Q \Leftrightarrow P \text{ and } Q \text{ have the same truth table} $$ + +23. Let $P$ be a set of parts shipped to a company from various suppliers. $S$ + is the relation defined on $P$ as follows: For every $x, y \in P$, + +$$ x S y \Leftrightarrow x \text{ has the same part number and is shipped from the same supplier as } y $$ + +24. Let $A$ be the set of identifiers in a computer program. It is common for + identifiers to be used for only a short part of the execution time of a + program and not to be used again to execute other parts of the program. In + such cases, arranging for identifiers to share memory locations makes + efficient use of a computer's memory capacity. Define a relation $R$ on $A$ + as follows: For all identifiers $x$ and $y$, + +$$ x R y \Leftrightarrow \text{ the values of } x \text{ and } y \text{ are stored in the same memory location during execution of the program} $$ + +25. $A$ is the "absolute value" relation defined on $\mathbb{R}$ as follows: + +$$ \text{For every } x, y \in \mathbb{R}, x A y \Leftrightarrow |x| = |y| $$ + +26. $D$ is the relation defined on $\mathbb{Z}$ as follows: For every + $m, n \in \mathbb{Z}$, + +$$ m D n \Leftrightarrow 3 | (m^2 - n^2) $$ + +27. $R$ is the relation defined on $\mathbb{Z}$ as follows: For every + $(m, n) \in \mathbb{Z}$, + +$$ m R n \Leftrightarrow 4 | (m^2 - n^2) $$ + +28. $I$ is the relation defined on $\mathbb{R}$ as follows: + +$$ \text{For every } x, y \in \mathbb{R}, m I n \Leftrightarrow x - y \text{ is an integer} $$ + +29. Define $P$ on the set $\mathbb{R} \times \mathbb{R}$ of ordered pairs of + real numbers as follows: For every + $(w, x), (y, z) \in \mathbb{R} \times \mathbb{R}$, + +$$ (w, x) P (y, z) \Leftrightarrow w = y $$ + +30. Define $Q$ on the set $\mathbb{R} \times \mathbb{R}$ as follows: For every + $(w, x), (y, z) \in \mathbb{R} \times \mathbb{R}$, + +$$ (w, x) Q (y, z) \Leftrightarrow x = z $$ + +31. Let $P$ be the set of all points in the Cartesian plane except the origin. + $R$ is the relation defined on $P$ as follows: For every $p_1$ and $p_2$ in + $P$, + +$$ p_1 R p_2 \Leftrightarrow p_1 \text{ and } p_2 \text{ lie on the same half-line emanating from the origin} $$ + +32. Let $A$ be the set of all straight lines in the Cartesian plane. Define a + relation $\mid \mid$ on $A$ as follows: For every $l_1$ and $l_2$ in $A$, + +$$ l_1 \mid \mid l_2 \Leftrightarrow l_1 \text{ is parallel to } l_2 $$ + +Then $\mid \mid$ is an equivalence relation on $A$. Describe the equivalence +classes of this relation. + +33. Let $A$ be the set of points in the rectangle with $x$ and $y$ coordinates + between $0$ and $1$. That is, + +$$ A = \{(x, y) \in \mathbb{R} \times \mathbb{R} | 0 \leq x \leq 1 \text{ and } 0 \leq y \leq 1\} $$ + +Define a relation $R$ on $A$ as follows: For all $(x_1, y_1) and $(x_2, y_2)$ in +$A$, + +$$ (x_1, y_1) R (x_2, y_2) \Leftrightarrow (x_1, y_1) = (x_2, y_2) $$ + +or + +$$ x_1 = 0 \text{ and } x_2 = 1 \text{ and } y_1 = y_2 $$ + +or + +$$ x_1 = 1 \text{ and } x_2 = 0 \text{ and } y_1 = y_2 $$ + +or + +$$ y_1 = 0 \text{ and } y_2 = 1 \text{ and } x_1 = x_2 $$ + +or + +$$ y_1 = 1 \text{ and } y_2 = 0 \text{ and } x_1 = x_2 $$ + +In other words, all points along the top edge of the rectangle are related to +the points along the bottom edge directly beneath them, and all points directly +opposite each other along the left and right edges are related to each other. +The points in the interior of the rectangle are not related to anything other +than themselves. Then $R$ is an equivalence relation on $A$. Imagine gluing +together all the points that are in the same equivalence class. Describe the +resulting figure. + +34. The documentation for the computer language Java recommends that when an + "equals method" is defined for an object, it be an equivalence relation. + That is, if $R$ is defined as follows: + +$$ x R y \Leftrightarrow \text{x.equals}(y) \text{ for all objects in the class} $$ + +then $R$ should be an equivalence relation. Suppose that in trying to optimize +some of the mathematics of a graphics application, a programmer creates an +object called a point, consisting of two coordinates in the plane. The +programmer defines an equals method as follows: If $p$ and $q$ are any points, +then + +$$ \text{p.equals}(q) \Leftrightarrow \text{ the distance from } p \text{ to } q \text{ is less than or equal to } c $$ + +where $c$ is a small positive number that depends on the resolution of the +computer display. Is the programmer's equals method an equivalence relation? +Justify your answer. + +35. Find an additional representative circuit for the input/output table of + Example 8.3.9. + +Let $R$ be an equivalence relation on a set $A$. Prove each of the statements in +36-41 directly from the definitions of equivalence relation and equivalence +class without using the results of Lemma 8.3.2, Lemma 8.3.3, or Theorem 8.3.4. + +36. For every $a$ in $a$, $a \in [a]$. + +37. For every $a$ and $b$ in $A$, if $b \in [a]$ then $a R b$. + +38. For every $a$, $b$, and $c$ in $A$, if $b R c$ and $c \in [a]$ then + $b \in [a]$. + +39. For every $a$ and $b$ in $A$, if $[a] = [b]$ then $a R b$. + +40. For every $a$, $b$, and $x$ in $A$, if $a R b$ and $x \in [a]$ then + $x \in [b]$. + +41. For every $a$ and $b$ in $A$, if $a \in [b]$ then $[a] = [b]$. + +42. Let $R$ be the relation defined in Example 8.3.12. + +a. Prove that $R$ is reflexive. + +b. Prove that $R$ is symmetric. + +c. List four distinct elements in $[(1, 3)]$. + +d. List four distinct elements in $[(2, 5)]$. + +43. In Example 8.3.12, define operations of addition $(+)$ and multiplication + $(\cdot)$ as follows: For every $(a, b), (c, d) \in A$, + +$$ [(a, b)] + [(c, d)] = [(ad + bc, bd)] $$ + +$$ [(a, b)] \cdot [(c, d)] = [(ac, bd)] $$ + +a. Prove that this addition is well defined. That is, show that if +$[(a, b)] = [(a', b')]$ and $[(c, d)] = [(c', d')]$, then +$[(ad + bc), bd] = [(a'd' + b'c', b'd')]$. + +b. Prove that this multiplication is well defined. That is, show that if +$[(a, b)] = [(a', b')]$ and $[(c, d)] = [(c', d')]$, then +$[(ac, bd)] = [(a'c', b'd')]$. + +c. Show that $[(0, 1)]$ is an identity element for addition. That is, show that +for any $(a, b) \in A$, + +$$ [(a, b)] + [(0, 1)] = [(0, 1)] + [(a, b)] = [(a, b)] $$ + +d. Find an identity element for multiplication. That is, find $(i, j)$ in $A$ so +that for every $(a, b)$ in $A$, +$[(a, b)] \cdot [(i, j)] = [(i, j)] \cdot [(a, b)] = [(a, b)]$. + +e. For any $(a, b) \in A$, show that $[(-a, b)]$ is an inverse for $[(a, b)]$ +for addition. That is, show that +$[(-a, b)] + [(a, b)] = [(a, b)] + [(-a, b)] = [(0, 1)]$. + +f. Given any $(a, b) \in A$ with $a \neq 0$, find an inverse for $[(a, b)]$ for +multiplication. That is, find $(c, d)$ in $A$ so that +$[(a, b)] \cdot [(c, d)] = [(c, d)] \cdot [(a, b)] = [(i, j)]$, where $[(i, j)]$ +is the identity element you found in part (d). + +44. Let $A = \mathbb{Z}^+ \times \mathbb{Z}^+$. Define a relation $R$ on $A$ as + follows: For every $(a, b)$ and $(c, d)$ in $A$, + +$$ (a, b) R (c, d) \Leftrightarrow a + d = c + b $$ + +a. Prove that $R$ is reflexive. + +b. Prove that $R$ is symmetric. + +c. Prove that $R$ is transitive. + +d. List five elements in $[(1, 1)]$. + +e. List five elements in $[(3, 1)]$. + +f. List five elements in $[(1, 2)]$. + +g. Describe the distinct equivalence classes of $R$. + +45. The following argument claims to prove that the requirement that an + equivalence relation be reflexive is redundant. In other words, it claims to + show that if a relation is symmetric and transitive, then it is reflexive. + Find the mistake in the argument. + +"**Proof:** Let $R$ be a relation on a set $A$ and suppose $R$ is symmetric and +transitive. For any two elements $x$ and $y$ in $A$, if $x R y$ then $y R x$ +since $R$ is symmetric. Thus it follows by transitivity that $x R x$, and hence +$R$ is reflexive." + +46. Let $R$ be a relation on a set $A$ and suppose $R$ is symmetric and + transitive. Prove the following: If for every $x$ in $A$ there is a $y$ in + $A$ such that $x R y$, then $R$ is an equivalence relation. + +47. Refer to the quote at the beginning of this section to answer the following + questions. + +a. What is the name of the Knight's song called? + +b. What is the name of the Knight's song? + +c. What is the Knight's song called? + +d. What _is_ the Knight's song? + +e. What is your (full, legal) name? + +f. What are you called? + +g. What _are_ you? (Do not answer this on paper; just think about it.) diff --git a/chapter_8/notes.md b/chapter_8/notes.md index 4fee6ce..bc8b5bf 100644 --- a/chapter_8/notes.md +++ b/chapter_8/notes.md @@ -96,3 +96,346 @@ the relation $R^t$ on $A$ that satisfies the following three properties: 3. If $S$ is any other transitive relation that contains $R$, then $R^t \subseteq S$. + +--- + +Page 529 + +**Definition** + +Given a partition of a set $A$, the **relation induced by the partition**, $R$, +is defined on $A$ as follows: For every $x, y \in A$, + +$$ x R y \Leftrightarrow \text{ there is a subset } A_i \text{ of the partition such that both } x \text{ and } y \text{ are in } A_i $$ + +--- + +Page 530 + +**Theorem 8.3.1** + +Let $A$ be a set with a partition and let $R$ be the relation induced by the +partition. Then $R$ is reflexive, symmetric, and transitive. + +**Proof:** + +Suppose $A$ is a set with a partition. In order to simplify notation, we assume +that the partition consists of only a finite number of sets. The proof for an +infinite partition is identical except for notation. Denote the partition +subsets by + +$$ A_1, A_2, \dots, A_n $$ + +Then $A_i \cap A_j = \emptyset$ whenever $i \neq j$, and +$A_1 \cup A_2 \cup A_3 \cdots \cup A_n = A$. The relation $R$ induced by the +partition is defined as follows: For every $x, y \in A$, + +$$ x R y \Leftrightarrow \text{ there is a set } A_i \text{ of the partition such that } x \in A_i \text{ and } y \in A_i $$ + +_[**Idea for the proof of reflexivity:** For $R$ to be reflexive means that each +element of $a$ is related by $R$ to itself. But by definition of $R$, for an +element $x$ to be related to itself means that $x$ is in the same subset of the +partition itself. Well, if $x$ is in some subset of the partition, then it is +certainly in the same subset as itself. And $x$ is in some subset of the +partition because the union of the subsets of the partition is all of $A$. This +reasoning is formalized as follows.]_ + +**Proof that $R$ is reflexive:** + +Suppose $x \in A$. Since $A_1, A_2, \dots A_n$ is a partition of $A$, it follows +that $x \in A_i$, for for some $i$, and so the statement + +there is a set $A_i$ of the partition such that $x \in A_i$ and $x \in A_i$ + +is true. Thus by definition of $R$, $x R x$. + +_[**Idea for the proof of symmetry:** For $R$ to be symmetric means that any +time one element is related to a second, then the second is related to the +first. Now for one element $x$ to be related to a second element $y$ means that +$x$ and $y$ are in the same subset of the partition. But if this is the case, +then $y$ is in the same subset of the partition as $x$, so $y$ is related to $x$ +by definition of $R$. This reasoning is formalized as follows.]_ + +**Proof that $R$ is symmetric:** + +Suppose $x$ and $y$ are elements of $A$ such that $x R y$. Then there is a +subset $A_i$ of the partition such that $x \in A_i$ and $y \in A_i$ by +definition of $R$. It follows that the statement + +there is a subset $A_i$ of the partition such that $y \in A_i$ and $x \in A_i$ + +is also true. Hence, by definition of $R$, $y R x$. + +_[**Idea for the proof of transitivity:** For $R$ to be transitive means that +any time one element of $A$ is related by $R$ to a second and that second is +related to a third, then the first element is related to the third. But for one +element to be related to another means that there is a subset of the partition +that contains both. So suppose $x$, $y$, and $z$ are elements such that $x$ is +in the same subset as $y$ and $y$ is in the same subset as $z$. Must $x$ be in +the same subset as $z$? Yes, because the subsets 9f the partition are mutually +disjoint. Since the subset that contains $x$ and $y$ has an element in common +with the subset that contains $y$ and $z$ (namely, $y$), the two subsets are +equal. But this means that $x$, $y$, and $z$ are all in the same subset, and so, +in particular, $x$ and $z$ are in the same subset. Hence $x$ is related by $R$ +to $z$. This reasoning is formalized as follows.]_ + +**Proof that $R$ is transitive:** + +Suppose $x$, $y$, and $z$ are in $A$ and $x R y$ and $y R z$. By definition of +$R$, there are subsets $A_i$ and $A_j$ of the partition such that + +$$ x \text{ and } y \text{ are in } A_i \quad \text{ and } \quad y \text{ and } z \text{ are in } A_j $$ + +Suppose $A_i \neq A_j$. _[We will deduce a contradiction.]_ Then +$A_i \cap A_j = \emptyset$ since $\{A_1, A_2, A_3, \dots, A_n\}$ is a partition +of $A$. But $y$ is in $A_i$ and $y$ is in $A_j$ also. Hence +$A_i \cap A_j \neq \emptyset$. _[This contradicts the statement that +$A_i \cap A_j = \emptyset$.]_ Thus $A_i = A_j$. It follows that $x$, $y$, and +$z$ are all in $A_i$, and so, in particular, + +$$ x \text{ and } z \text{ are in } A_i $$ + +Thus $x R z$ by definition of $R$. + +--- + +Page 531 + +**Definition** + +Let $A$ be a set and $R$ a relation on $A$. $R$ is an **equivalence relation** +if, and only if, $R$ is reflexive, symmetric, and transitive. + +--- + +Page 533 + +**Definition** + +Suppose $A$ is a set and $R$ is an equivalence relation on $A$. For each element +$a$ in $A$, the **equivalence class of $a$**, denoted $[a]$ and called the +**class of $a$** for short, is the set of all elements $x$ in $A$ such that $x$ +is related to $a$ by $R$. + +In symbols: + +$$ [a] = \{x \in A | x R a\} $$ + +--- + +Page 536 + +**Lemma 8.3.2** + +Suppose $A$ is a set, $R$ is an equivalence relation on $A$, and $a$ and $b$ are +elements of $A$. If $a R b$, then $[a] = [b]$. + +--- + +Page 536 + +**Proof of Lemma 8.3.2** + +Let $A$ be a set, let $R$ be an equivalence relation on $A$, and suppose + +$$ a \text{ and } b \text{ are elements of } A \text{ such that } a R b $$ + +_[We must show that $[a] = [b]$.]_ + +**Proof that $[a] \subseteq [b]$:** + +Let $x \in [a]$. _[We must show that $x \in [b]$.]_ + +Since + +$$ x \in [a] $$ + +then + +$$ x R a $$ + +by definition of class. But + +$$ a R b $$ + +by hypothesis. Thus, by transitivity of $R$, + +$$ x R b $$ + +Hence + +$$ x \in [b] $$ + +by definition of class. _[This is what was to be shown.]_ + +**Proof that $[b] \subseteq [a]$. + +Let $x \in [b]$. _[We must show that $x \in [a]$.]_ + +Since + +$$ x \in [b] $$ + +then + +$$ x R b $$ + +by definition of class. Now + +$$ a R b $$ + +by hypothesis. Thus, since $R$ is symmetric, + +$$ b R a $$ + +also. Then, since $R$ is transitive and $x R b$ and $b R a$, + +$$ x R a $$ + +Hence, + +$$ x \in [a] $$ + +by definition of class. _[This is what was to be shown.]_ + +Since $[a] \subseteq [b]$ and $[b] \subseteq [a]$, it follows that $[a] = [b]$ +by definition of set equality. + +--- + +Page 537 + +**Lemma 8.3.3** + +If $A$ is a set, $R$ is an equivalence relation on $A$, and $a$ and $b$ are +elements of $A$, then + +$$ \text{either } [a] \cap [b] = \emptyset \quad \text{ or } \quad [a] = [b] $$ + +--- + +Page 537 + +**Proof of Lemma 8.3.3** + +Suppose $A$ is a set, $R$ is an equivalence relation on $A$, $a$ and $b$ are +elements of $A$, and + +$$ [a] \cap [b] \neq \emptyset $$ + +_[We must show that $[a] = [b]$.]_ + +Since $[a] \cap [b] \neq \emptyset$, there exists an element $x$ in $A$ such +that $x \in [a] \cap [b]$. By definition of intersection, + +$$ x \in [a] \quad \text{ and } \quad x \in [b]$$ + +and so + +$$ x R a \quad \text{ and } \quad x R b $$ + +by definition of class. Since $R$ is symmetric _[being an equivalence relation]_ +and $x R a$, then $a R x$. But $R$ is also transitive _[since it is an +equivalence relation]_, and so, since $a R x$ and $x R b$, + +$$ a R b $$ + +Now $A$ and $b$ satisfy the hypothesis of Lemma 8.3.2. Hence, by that lemma, + +$$ [a] = [b] $$ + +_[as was to be shown]._ + +--- + +Page 537 + +**Theorem 8.3.4 The Partition Induced by an Equivalence Relation** + +If $A$ is a set and $R$ is an equivalence relation on $A$, then the distinct +equivalence classes of $R$ form a partition of $A$; that is, the union of the +equivalence classes is all of $A$, and the intersection of any two distinct +classes is empty. + +--- + +Page 538 + +**Proof of Theorem 8.3.4** + +Suppose $A$ is a set and $R$ is an equivalence relation on $A$. For notational +simplicity, we assume that $R$ has only a finite number of distinct equivalence +classes, which we denote + +$$ A_1, A_2, \dots, A_n $$ + +where $n$ is a positive integer. (When the number of classes is infinite, the +proof is identical except for notation.) + +**Proof that $A = A_1 \cup A_2 \cup \cdots \cup A_n$:** + +_[We must show that $A \subseteq A_1 \cup A_2 \cup \cdots \cup A_n$ and that +$A_1 \cup A_2 \cup \cdots \cup A_n \subseteq A$.]_ + +To show that $A \subseteq A_1 \cup A_2 \cup \cdots \cup A_n$, suppose $x$ is any +element of $A$. _[We must show that $x \in A_1 \cup A_2 \cup \cdots A_n$.]_ By +reflexivity of $R$, $x R x$. And this implies that $x \in [x]$ by definition of +class. Since $x$ is in _some_ equivalence class, it must be in one of the +distinct equivalence classes $A_1, A_2, \dots$, or $A_n$. Thus $x \in A_i$ for +some index $i$, and hence $x \in A_1 \cup A_2 \cup \cdots \cup A_n$ by +definition of union _[as was to be shown]_. + +To show that $A_1 \cup A_2 \cup \cdots \cup A_n \subseteq A$, suppose +$x \in A_1 \cup A_2 \cup \cdots \cup A_n$. _[We must show that $x \in A$.]_ Then +$x \in A_i$ for some $i = 1, 2, \dots, n$, by definition of union. Now each +$A_i$ is an equivalence class of $R$, and equivalence classes are subsets of +$A$. Hence $A_i \subseteq A$ and so $x \in A$ _[as was to be shown]._ + +Since $A \subseteq A_1 \cup A_2 \cup \cdots A_n$ and +$A_1 \cup A_2 \cup \cdots \cup A_n \subseteq A$, then by definition of set +equality, $A = A_1 \cup A_2 \cup \cdots \cup A_n$. + +**Proof that the distinct classes of $R$ are mutually disjoint:** + +Suppose that $A_i$ and $A_j$ are any two distinct equivalence classes of $R$. +_[We must show that $A_i$ and $A_j$ are disjoint.]_ Since $A_i$ and $A_j$ are +distinct, then $A_i \neq A_j$. And since $A_i$ and $A_j$ are equivalence classes +of $R$, there must exist elements $a$ and $b$ in $A$ such that $A_i = [a]$ and +$A_j = [b]$. + +By Lemma 8.3.3, + +$$ \text{either } [a] \cap [b] = \emptyset \quad \text{ or } \quad [a] = [b]$$ + +Now $[a] \neq [b]$ because $A_i \neq A_j$, and hence $[a] \cap [b] = \emptyset$. +Thus $A_i \cap A_j = \emptyset$, and so $A_i$ and $A_j$ are disjoint _[as was to +be shown]._ + +--- + +Page 540 + +**Definition** + +Suppose $R$ is an equivalence relation on a set $A$ and $S$ is an equivalence +class of $R$. A **representative** of the class $S$ is any element $a$ such that +$[a] = S$. + +-- + +Page 541 + +**Definition** + +Let $m$ and $n$ be integers and let $d$ be a positive integer. We say that **$m$ +is congruent to $n$ modulo $d$** and write + +$$ m = n (\mod d) $$ + +if, and only if, + +$$ d | (m - n) $$ + +Symbolically: + +$$ m \equiv n(\mod d) \Leftrightarrow d | (m - n) $$ diff --git a/chapter_8/test_yourself.md b/chapter_8/test_yourself.md index be86a2e..b0bb37d 100644 --- a/chapter_8/test_yourself.md +++ b/chapter_8/test_yourself.md @@ -77,3 +77,27 @@ $\exists x, y, z \in A, (x R y \wedge y R z) \to x \cancel{R} z$ $R^t$ is transitive; $R \subseteq R^t$; if $S$ is any other transitive relation that contains $R$, then $R^t \subseteq S$ + +--- + +Page 543 + +**Test Yourself** + +1. For a relation on a set to be an equivalence relation, it must be ____. + +2. The notation $m \equiv n (\mod d)$ is read "____" and means that ____. + +3. Given an equivalence relation $R$ on a set $A$ and given an element $a$ in + $A$, the equivalence class of $a$ is denoted ____ and is defined to be ____. + +4. If $A$ is a set, $R$ is an equivalence relation on $A$, and $a$ and $b$ are + elements of $A$, then either $[a] = [b]$ or ____. + +5. If $A$ is a set and $R$ is an equivalence relation on $A$, then the distinct + equivalence classes of $R$ form ____. + +6. Let $A = \mathbb{Z} \times (\mathbb{Z} - \{0\})$, and define a relation $R$ + on $A$ by specifying that for every $(a, b)$ and $(c, d)$ in $A$, + $(a, b) R (c, d)$ if, and only if, $ad = bc$. Then there is exactly one + equivalence class of $R$ for each ____.