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chapter_8/exercises.md
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@ -1957,3 +1957,405 @@ Omitted.
$$ A = \{a[1], a[2], \dots, a[n]\} $$ $$ A = \{a[1], a[2], \dots, a[n]\} $$
Omitted. Omitted.
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**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.)

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chapter_8/notes.md
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@ -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 3. If $S$ is any other transitive relation that contains $R$, then
$R^t \subseteq S$. $R^t \subseteq S$.
---
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:
$$ m \equiv n(\mod d) \Leftrightarrow d | (m - n) $$

24
chapter_8/test_yourself.md
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@ -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 $R^t$ is transitive; $R \subseteq R^t$; if $S$ is any other transitive relation
that contains $R$, then $R^t \subseteq S$ 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 ____.