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Logic, sets, and recursion

Robert L. Causey

Chapter 2

Basic Set Theory - all with Video Answers

Educators

Chapter Questions

03:04

Problem 1

Let $A=|a, b|, B=\{b, c \mid$, where $a, b, c$ are mutually distinct. Let the domain of discourse be $A \cup B$. Write the elements of each of the following sets in braces.
$$
A \cap B
$$
$B-A$
$B^{\prime}$
$(A \cap B)^{\prime}$
$(A-B) \cup(B-A)$
$A-B$
$A^{\prime}$
$(B-A)^{\prime}$
$(A \cup B)^{\prime}$
$(A \cup B)-(A \cap B)$

Doruk Isik
Doruk Isik
Numerade Educator
01:19

Problem 2

Complete the proof of De Morgan's Theorem, Theorem 2-11.

Nick Johnson
Nick Johnson
Numerade Educator
02:11

Problem 3

Complete the proof of Theorem 2-14.

James Kiss
James Kiss
Numerade Educator
02:04

Problem 4

Theorem. For any sets, $A, B, C, D$, if $A \subseteq B \& C \subseteq D$, then $A \cup C \subseteq B \cup D$.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:18

Problem 5

Theorem. For any sets $A, B$,
$$
(A-B)=A \cap B^{\prime}
$$
and
$$
(A-B)^{\prime}=A^{\prime} \cup B .
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator

Problem 6

For any sets $A, B, C, A-(A-B)=A \cap B$.

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04:00

Problem 7

Let $A, B, C$ be sets, then
$$
(A-B) \cup C=(C \cup A) \cap\left(C \cup B^{\prime}\right) .
$$

Doruk Isik
Doruk Isik
Numerade Educator
03:04

Problem 8

For any sets $A, B$,
$$
(A-B) \cup(B-A)=(A \cup B)-(A \cap B) .
$$

Doruk Isik
Doruk Isik
Numerade Educator
03:04

Problem 9

Theorem. For any sets $A, B$,
$$
A=B \text { iff }((A \cup B) \subseteq(A \cap B)) .
$$

Doruk Isik
Doruk Isik
Numerade Educator

Problem 10

For any sets $A, B, A \subseteq B$ iff $B^{\prime} \subseteq A^{\prime}$.

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Problem 11

For any sets $A, B, C$,
$$
(A \cap B) \subseteq C
$$
iff
$$
(A \cap B) \subseteq(B \cap C) \text { and }(A \cap B) \subseteq(A \cap C) .
$$

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00:56

Problem 12

Let
$$
M=\{0,1,2,11,3\}, N=\{7,1,3,2\} \text {, and } S=\{8,3,0,2,11\} .
$$
Also let $A=|M, N, S|$. What is $\cup A$ ? What is $\cap A$ ?

Gaurav Kalra
Gaurav Kalra
Numerade Educator
00:19

Problem 13

Let $S, T$ be any sets. Then $S=\cap|S|$ and $S \cap T=\cap \mid S, T\}$.

Amy Jiang
Amy Jiang
Numerade Educator
01:48

Problem 14

Let $S, T$ be any sets. Then what is $\cup\{S \mid$ and what is $\cup\{S, T\}$ ? Carefully state and prove your answers. [Hint. Compare with the previous exercise.]

Nick Johnson
Nick Johnson
Numerade Educator

Problem 15

For any set $A, A=\cup P(A)$. [Hint. If $x \in A$, then $|x| \subseteq A$.]

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Problem 16

For any sets, $A, B$,
$$
\mathcal{P}(A \cap B)=P(A) \cap \mathcal{P}(B) .
$$

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01:07

Problem 17

For any sets, $A, B, A \subseteq B$ iff $P(A) \subseteq P(B)$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:25

Problem 18

Let the domain of discourse be the real numbers, $R e$. Four relations are defined as follows, where $x, y \in R e$,
$$
\begin{aligned}
& P x y \text { iff } y=x^2-2, \\
& \text { Lxy iff } y=2 x+1, \\
& R x y \text { iff } y \geq x^2-2, \\
& S x y \text { iff } y \leq 2 x+1 .
\end{aligned}
$$
1. Sketch a plot of each of these relations on the Cartesian coordinate system. Use different shadings or colors to indicate regions of the plane.
2. What are the elements (ordered pairs) of the relation $P \cup L$ ?
3. What are the elements of the relation $P \cap L$ ? [This requires solving a simple quadratic equation.]
4. Using set abstraction notation, write a definition for the relation $R \cap S$. On your plot, represent $R \cap S$ with special shading or coloration.

AG
Ankit Gupta
Numerade Educator
04:28

Problem 19

The Omega Corporation is described earlier in the text. Using the information in the text and in Figure 2-4, list all of the elements of the sets $M, R, S, P$. Display the elements of $D$ in a relation table.

Mj Santos
Mj Santos
Numerade Educator
02:02

Problem 20

Let $A \neq \varnothing$ be a set, and let $I=\{\langle x, x\rangle \mid x \in A\}$. Then $I \subseteq P(P(A))$. [Hint. Use Definition 2-15 of an ordered pair.]

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:46

Problem 21

Let $A, B, C$ be nonempty sets. If $A \subseteq B$, then
$$
(A \times C) \subseteq(B \times C) .
$$

Doruk Isik
Doruk Isik
Numerade Educator
00:59

Problem 22

Let $A, B, C$ be nonempty sets. Then
$$
A \times(B \cap C)=(A \times B) \cap(A \times C) .
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
06:24

Problem 23

Let $A, B, C$ be nonempty sets. Then
$$
A \times(B-C)=(A \times B) \cap\left(A \times C^{\prime}\right) .
$$

Mengchun Cai
Mengchun Cai
Numerade Educator

Problem 24

Let $R, S \subseteq(A \times B)$ be nonempty binary relations. Then
$$
(R \cup S)^{-1}=\left(R^{-1} \cup S^{-1}\right)
$$
and
$$
(R \cap S)^{-1}=\left(R^{-1} \cap S^{-1}\right) .
$$

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01:29

Problem 25

Suppose we have information about some cars that are denoted by their license plate numbers. There are two sets of data represented by binary relations as follows:
$C=1<223 \mathrm{ABC}$, red $>,<105 \mathrm{XJQ}$, blue $>,<804 \mathrm{EWV}$, green $>$, $<831 \mathrm{EOZ}$, blue>, <834BTJ, red>, <979XPT, yellow $>$, <333NJL, green $>,<113 \mathrm{NPO}$, black $>,<660 \mathrm{AAA}$, white $>1$
$S=1<223 \mathrm{ABC}$, compact $>,<105 \mathrm{XJQ}$, midsize $>$, <804EWV, compact $>$, <834BTJ, large $>$, <054HHH, midsize $>$, <979XPT, large $>$, $<333 \mathrm{NJL}$, compact $>,<113 \mathrm{NPO}$, compact $>$ |A new 3-ary relation $J$ is defined as follows: For any license number $n$, color $x$, and size $y$, Jnxy iff ( $C n x$ and $S n y$ ).
1. Using the above data, find the set of all triples in $J$. Exhibit these triples in a relation table with column headings, license number, color, size.
2. A new relation $R$ is defined by $R y x n$ iff Jnxy. Make a table that exhibits all triples of $R$ of the form, <compact, $x, n>$.
3. Find the license numbers of all cars (if any) that are compact and either red or green.

James Kiss
James Kiss
Numerade Educator
02:25

Problem 26

Give an example of a binary relation that is connected on a set $A$, but not strongly connected on $A$.

Mj Santos
Mj Santos
Numerade Educator
05:40

Problem 27

Let $R$ be a binary relation on nonempty set $A$, and let $R^{-1}$ be the converse of $R$. Then:
1. If $R$ is asymmetric on $A$, then $R^{-1}$ is asymmetric on $A$.
2. If $R$ is irreflexive on $A$, then $R^{-1}$ is irreflexive on $A$.
3. If $R$ is transitive on $A$, then $R^{-1}$ is transitive on $A$.
4. If $R$ is symmetric on $A$, then $R^{-1}$ is symmetric on $A$.

Nidhi Singhi
Nidhi Singhi
Numerade Educator
03:08

Problem 28

Let $R$ be a binary relation on $A \neq \varnothing$. If $R$ is transitive and irreflexive on $A$, then $R$ is asymmetric on $A$.

Mj Santos
Mj Santos
Numerade Educator
00:54

Problem 29

Let $R$ be a strongly connected relation on $A \neq \varnothing$. Then:
1. $R$ is reflexive on $A$.
2. $R^{-1}$ is also strongly connected on $A$.
3. $R$ is connected on $A$.

James Chok
James Chok
Numerade Educator

Problem 30

Let $\Gamma$ be the set of all $S C$ sentences, and, for any $\phi, \psi \in \Gamma$, define $T \phi \psi$ iff $\phi$ and $\psi$ are tautologically equivalent. Prove that $T$ is an equivalence relation on $\Gamma$.

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Problem 31

Let $x, y, u, v \in \mathrm{Nat}^{+}$. Define $R$ by: $R\langle x, y\rangle\langle u, v\rangle$ iff $x v=y u$, where $x v$ and $y u$ are products. Prove that $R$ is an equivalence relation on $\mathrm{Nat}^{+} \times \mathrm{Nat}^{+}$.

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Problem 32

If $R$ and $S$ are equivalence relations on set $A \neq \varnothing$, then $R \cap S$ is an equivalence relation on $A$.

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06:06

Problem 33

Give an example of a binary relation $R$ on a set $A$, such that $R$ is strongly connected on $A$ and is also transitive on $A$, but $R$ is not an equivalence relation on $A$. Show that your example relation is not an equivalence relation on $A$.

Abigail Martyr
Abigail Martyr
Numerade Educator

Problem 34

Let $R$ be a binary relation on $A . R$ is countertransitive if and only if for every $x, y, z \in A$,
$$
\text { if }(R x y \& R y z) \text {, then } R z x .
$$
Theorem. $R$ is an equivalence relation on $A$ iff $R$ is reflexive and countertransitive on $A$.

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03:00

Problem 35

Suppose one has a domain $D$ of toy building blocks with the following properties: There are exactly two disjoint sizes, big, $B$, and little, $L$, and exactly three disjoint colors, purple, $P$, red, $R$, and green, $G$. Define two relations on $D$ by:
$$
S x y \text { iff }((B x \& B y) \text { or }(L x \& L y))
$$
$C x y$ iff $((P x \& P y)$ or $(R x \& R y)$ or $(G x \& G y))$.
1. Briefly describe the meanings of $S$ and $C$.
2. Prove that $S$ and $C$ are equivalence relations on $D$.
3. Prove that $S \cap C$ is an equivalence relation on $D$.
4. Briefly describe the kinds of partition cells that are generated by $S \cap C$. What is the maximum number of cells that might occur? Why?

Nick Johnson
Nick Johnson
Numerade Educator
03:49

Problem 36

Let $S$ be a set and $P$ be a nonempty set of nonempty subsets of $S$. Define $E_P$ on $S$ as follows: For any $x, y \in S$,
$$
E_P x y \text { iff ( for every } A \in P, x \in A \text { iff } y \in A \text { ). }
$$
1. Describe an example in which $S$ is a set of at least ten uniformly colored marbles, and in which $P$ has nonempty sets for red, green, blue, large, and small marbles. Large and small are disjoint, and any two different colors are disjoint. Describe in words the meaning of $E_P$ for your example. Show that $E_P$ partitions $S$, and describe the cells of this partition.
2. Let $S$ and $P$ be any sets satisfying the preceding conditions. Prove that $E_P$ is an equivalence relation on $\mathrm{S}$. [Remark. This theorem yields a very general method for classifying the elements of $S$ according to the $P$-subsets they have in common. Two elements of $S$ are of the same type or kind iff they fall into exactly the same $P$-subsets of $S$. See Causey (1977), Chapter 3, for generalizations and applications.]

James Kiss
James Kiss
Numerade Educator

Problem 37

If $\langle A, R\rangle$ is a strict simple order, then $\left\langle A, R^{-1}\right\rangle$ is also a strict simple order.

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Problem 38

If $\langle A, R\rangle$ is a partial order, then $\left\langle A, R^{-1}\right\rangle$ is a partial order.

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Problem 39

Define $R$ on Nat $\times$ Nat as follows: For any $x, y, u, v$ in Nat,
$$
R<x, y><u, v\rangle \text { iff }(x \leq u \& y \leq v) .
$$
$R$ is a binary relation between pairs. It might help to think of $R$ as a relation between points on the Cartesian coordinate system; then you can draw a sketch of how it appears.
1. Prove that $R$ is a partial order on $\mathrm{Nat} \times \mathrm{Nat}$.
2. Give a counterexample to the assertion that $R$ is an equivalence relation on Nat $\times$ Nat.
3. Give a counterexample to the assertion that $R$ is a simple order on Nat $\times$ Nat.

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01:29

Problem 40

Let $\langle A, L\rangle$ be a strict simple order. Then:
1. $L$ is irreflexive on $\mathrm{A}$.
2. For any $x, y, z \in A$, one and only one of the following holds: $x=y$, Lxy, Lyx.
[Remark. The second statement is called the trichotomy law. The fact that $L$ is asymmetric, and hence irreflexive, together with connectedness, will help to prove the trichotomy law.]

Doruk Isik
Doruk Isik
Numerade Educator
01:02

Problem 41

Let $\langle A, L\rangle$ and $\langle A, M\rangle$ be strict simple orders with $L \subseteq M$. Then $L=M$.

Megan Mcfarland
Megan Mcfarland
Numerade Educator

Problem 42

Let $A=\{a, b, c, \cdots, x, y, z\}$ be the set ofters in the English alphabet. Let $P$ (precedes) be the usual alphabetical ordering of these letters, so that $P a b, P b c, P c g, P d q, \cdots$, etc. Notice that $\langle A, P\rangle$ is a strict simple order. Now let $W=|\alpha \beta| \alpha, \beta \in A \mid$, where $\alpha \beta$ denotes the result of writing the letter $\alpha$ followed by the letter $\beta$. Thus, $W$ is the set of all two letter "words" (strings), such as 'ab', 'ee', 'zx', etc. Now define $L$ on $W$ as follows: if word $_1$, word $_2 \in W$, and word $_1=\alpha \beta$ and word $_2=\gamma \delta$, then
Lword $_1$ word $_2$ iff $(P \alpha \gamma$ or $(\alpha=\gamma \quad \& \quad P \beta \delta))$.
Use the $L$ relation to sort the following list of two-letter "words" into alphabetical ordering: $\mathrm{lm}$, ma, an, bk, zz, ab, bx, ji, mn, it, to, aa, ij, xb, ik, $\mathrm{ji}, \mathrm{zy}, \mathrm{ac}, \mathrm{mm}$. Notice the next exercise.

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08:12

Problem 43

Let $A, P, W$, and $L$ be the same as in the preceding exercise. Prove that $\langle W, L\rangle$ is a strict simple order. [Remark. The proof requires considering several cases, so please have patience with it. $L$ is called the lexicographic ordering of $W$. It can be generalized to words with $n$ letters. This ordering is widely used in mathematics and computing theory. It is also used to order the words in dictionaries, phone directories, etc. I

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:46

Problem 44

Let $\left\langle A,<_A\right\rangle$ and $\left\langle B,<_B\right\rangle$ be strict simple orders. Define $L$ on $A \times B$ as follows: For any $\langle x, y\rangle$ and $\langle u, v\rangle$ in $A \times B$,
$$
L<x, y><u, v\rangle \text { iff }\left(x<_A u \text { or }\left(x=u \& y<_B v\right)\right) \text {. }
$$
Then $\langle A \times B, L\rangle$ is a strict simple order. [Remark. This theorem is an abstract generalization of the previous exercise. For example, suppose that we have two ordered sequences (or arrays) of numbers. This theorem provides one method for ordering their Cartesian product.]

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 45

Let $\langle D, Q\rangle$ be a relational system in which $Q$ is reflexive and transitive on $D$. Such a system is called a quasiorder. Every partial order is a quasi-order, but in addition, partial orders are antisymmetric. Since they are not required to be antisymmetric, quasi-orders are more convenient for representing some kinds of data than are partial orders. In psychological and economic theories, a quasiorder is sometimes used to represent a person's preferences among a set of choices.
Given a quasi-order, we can define, for $x, y \in D, I x y$ iff ( $Q x y \& Q y x)$. The relation $I$ is called the indifference relation of $Q$.
1. Use this definition of $I$ together with the reflexivity and transitivity of $Q$ to prove that $I$ is an equivalence relation on $D$.
2. Suppose a restaurant has nine items on its menu, $d_1, \cdots, d_9$. Let $P x y$ represent that Bob prefers menu item $y$ to item $x$. Bob expresses the following preferences: $P d_1 d_2, P d_2 d_4, P d_4 d_2, P d_2 d_3, P d_3 d_5, P d_5 d_7$, $P d_7 d_5, P d_6 d_7, P d_6 d_8, P d_8 d_6, P d_6 d_9$. In addition to this data, the only other information we have is that $P$ is a quasi-order. What are the equivalence classes corresponding to $I$ ? Which, if any, is true: $P d_4 d_6, I d_4 d_6$, or $P d_6 d_4$ ? Which, if any, is true: $P d_1 d_7, I d_1 d_7, P d_7 d_1$ ? Give reasons for your answers. Also, sketch the graph of Bob's preferences, and indicate the equivalence classes on this graph.

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01:00

Problem 46

Consider the relational system $\left\langle R e^{+}, f\right\rangle$, where $f=\{\langle x, y\rangle|x y=7|$, where $x y$ is the product of $x$ and $y$.
1. Prove that $f$ is a function.
2. Now introduce functional notation by: $y=f(x)$ iff $<x, y>\in f$. Prove that $f$ is one-one.
3. Write an expression for the inverse of $f$; show that it is the inverse.

Amy Jiang
Amy Jiang
Numerade Educator
05:35

Problem 47

If $r \in R a t$, then $r$ can be expressed as a rational fraction, $x / y$, where $x, y \in$ Int. For example, $-1 / 2$ equals $1 /-2,-2 / 4,3 /-6$, etc. Let $x, y \in$ Int with $y \neq 0$. Let $x / y$ be the corresponding rational fraction. For each of the following, determine whether or not the relation $f$ is a function from $R a t$ to $R e$. If it is a function, then also determine whether or not it is one-one. Use elementary algebra (including the laws of exponents) to give proofs or counterexamples.
1. $f x / y u$ iff $u=x^2 / y$.
2. $f x / y u$ iff $u=x^2 / y^2$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:33

Problem 48

For $x, y \in R e$, let $m(x, y)=m(<x, y\rangle)=x y$ be the product of $x$ and $y$, i.e., $x$ times $y$. Define the following functions on $R e: f(x)=x+1$, $g(x)=x-1, h(x)=m(f(x), g(x))$. Prove or disprove each of the following:
$f$ is one-one on Re. $f$ is onto $R e$. $g$ is one-one on Re. $g$ is onto Re.
$h$ is one-one on Re. $h$ is onto Re.

Nick Johnson
Nick Johnson
Numerade Educator
01:00

Problem 49

Briefly state what the error is in this alleged "proof." Assume the elementary properties of the integers.
Let $f$ be the function from the integers to the integers satisfying, for any integer $x, f(x)=x^2-16$. Then
$$
f(4)=f(-4)=0 .
$$
Let $f^{-1}$ be the inverse of $f$. Then
$$
f^{-1} \circ f(4)=f^{-1} \circ f(-4) .
$$
Hence,
$$
4=-4 .
$$

Amy Jiang
Amy Jiang
Numerade Educator
03:04

Problem 50

Let $A, B, C$ be nonempty sets and suppose that
$$
f: A \rightarrow C
$$
and
$$
f: B \rightarrow C .
$$
Then $\operatorname{Img}(f,(A \cup B))=\operatorname{Img}(f, A) \cup \operatorname{Img}(f, B)$.

Doruk Isik
Doruk Isik
Numerade Educator
07:07

Problem 51

Let $A$ and $B$ be nonempty sets, and suppose that
$$
f: A \rightarrow C
$$
and
$$
f: B \rightarrow C .
$$
1. Prove that $\operatorname{Img}(f,(A \cap B)) \subseteq \operatorname{Img}(f, A) \cap \operatorname{Img}(f, B)$.
2. Show by an example that it is possible to have
$$
\operatorname{Img}(f,(A \cap B)) \neq \operatorname{Img}(f, A) \cap \operatorname{Img}(f, B) .
$$
3. Prove that, if $f$ is one-one, then
$$
\operatorname{Img}(f, A) \cap \operatorname{Img}(f, B) \subseteq \operatorname{Img}(f,(A \cap B)) .
$$

James Chok
James Chok
Numerade Educator
01:01

Problem 52

Define $f(x, y)=f(\langle x, y\rangle)=2^x 3^y$, for $x, y \in$ Nat. Assume elementary facts of arithmetic and algebra (including properties of exponents) to prove that $f$ maps Nat $\times$ Nat one-one to Nat.

Raj Bala
Raj Bala
Numerade Educator

Problem 53

Let $A=\{1,2,3,4\}$. A function that maps $A$ one-one and onto $A$ is called a permutation of $A$. The identity function, $i_A$, is a permutation, and there are many others, e.g., the function $p$ such that $p(1)=2$, $p(2)=1$, and $p(x)=x$, for $x=3,4$. Prove that the composition of any two permutations of $A$ is also a permutation of $A$.

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02:55

Problem 54

Let $f$ be a one-one function from $R e^{+}$onto $R e^{+}$. Define
$$
g: \operatorname{Re}^{+} \rightarrow \operatorname{Re}^{+}
$$
by: for any $x \in R e^{+}, g(x)=1 / f(x)$. Prove that $g$ is also one-one and that it maps onto $\mathrm{Re}^{+}$. You may assume the elementary arithmetical and algebraic properties of the real numbers.

Clarissa Noh
Clarissa Noh
Numerade Educator
02:10

Problem 55

Let $A, B$ be nonempty sets, and suppose that
$$
f: A \rightarrow B,
$$
and that $f$ is one-one on $A$. Let $C$ be a nonempty subset of $A$. Then
$$
C=\operatorname{Img}\left(f^{-1}, \operatorname{Img}(f, C)\right) .
$$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:20

Problem 56

Let $A$ be a nonempty set and let $f, g$ be one-one functions from $A$ onto $A$. Then $g \circ f$ is one-one and $(g \circ f)^{-1}=f^{-1} \circ g^{-1}$.

Madi Sousa
Madi Sousa
Numerade Educator
14:32

Problem 57

Let $\langle A, L\rangle$ be a relational system such that, for any $x, y \in A$, one and only one of the following holds: $x=y$, $L x y, L y x$. Let $f: A \rightarrow A$ such that, for any $x, y \in A$, if $L x y$, then $L f(x) f(y)$. Then $f$ maps $A$ one-one to $A$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:22

Problem 58

Let $f: R e \rightarrow R e . f$ is strictly increasing iff for any $x, y \in R e$, if $x<y$, then $f(x)<f(y)$. Assume that $<$ is a strict simple order on $R e$, and prove the following.

Theorem. If $f$ is a strictly increasing function from $R e$ to $R e$, then $f$ is one-one on $R e$, and $f^{-1}$ is strictly increasing on $R e$. [Hint. Look at E2.40 and E2.57.]

Helen Latting
Helen Latting
Numerade Educator
01:49

Problem 59

Let $A, L$, and $f$ satisfy the antecedent conditions of the theorem in E2.57. Give a counterexample to show that $f$ need not map onto $A$.

Will Erickson
Will Erickson
Numerade Educator

Problem 60

Let $A$ be a nonempty set and let $f: A \rightarrow A$ be one-one and onto $A$. Let $B$ be any nonempty subset of $A$. Define the function $f_B$ by:
$$
f_B=|<x, y>| x \in B \& y=f(x) \mid .
$$
$f_B$ is called the restriction of $f$ to $B$. Prove, or give a counterexample to, each of the following:
1. $f_B$ is a one-one function on its domain $B$.
2. $f_B$ maps $B$ onto $B$.

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03:23

Problem 61

Let $A, B$ be nonempty sets and $f: A \longrightarrow B$. Define $R$ on $A$ by: $R x y$ iff $f(x)=f(y)$.
1. Prove that $R$ is an equivalence relation on $A$.
2. Let $a \in A$ and $[a]$ be the $R$-equivalence class of $a$. Prove that all elements in $[a]$ have the same $f$ value, i.e., $f$ maps every element in $[a]$ into exactly one element of $B$.
3. Suppose that it is proposed to define a new function
$$
F: A / R \rightarrow B
$$
by: For any $x \in A$,
$$
F([x])=f(x) .
$$
Prove that $F$ really is a function.
4. In the special case where $f$ is one-one, prove that every cell of $A / R$ has exactly one member.

James Stillman
James Stillman
Numerade Educator
14:32

Problem 62

Let $\langle A, L\rangle,\langle B, R\rangle$ be relational systems in which $L$ and $R$ are binary relations. Let $f: A \rightarrow B$, where $f$ is one-one and maps onto $B$. Suppose that, for every $x, y \in A$,
Lxy iff $R f(x) f(y)$.
Then: if $L$ is a strict simple order on $A$, then $R$ is a strict simple order on B.
[Remark. $\langle A, L\rangle$ is said to be isomorphic to $\langle B, R\rangle$ when there is a one-one function $f$ mapping $A$ onto $B$ satisfying the above biconditional. Such a one-one function mapping $A$ onto $B$ is said to preserve the ordering relations and such a function is called an isomorphism. Isomorphisms, and other kinds of related mappings, play an important role in much of modern mathematics.]

Anthony Ramos
Anthony Ramos
Numerade Educator