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Chapter 9

Relations

Educators


Problem 1

List the ordered pairs in the relation $R$ from $A=\{0,1,2,3,4\}$ to $B=\{0,1,2,3\},$ where $(a, b) \in R$ if and only if
$$
\begin{array}{ll}{\text { a) } a=b .} & {\text { b) } a+b=4} \\ {\text { c) } a>b .} & {\text { d) } a | b} \\ {\text { e) } \operatorname{gcd}(a, b)=1 .} & {\text { f) } \operatorname{lcm}(a, b)=2}\end{array}
$$

James C.
Numerade Educator

Problem 2

a) List all the ordered pairs in the relation $R=\{(a, b) | a \text { divides } b\}$ on the set $\{1,2,3,4,5,6\} .$
b) Display this relation graphically, as was done in Example $4 .$
c) Display this relation in tabular form, as was done in Example 4.

Mj S.
Numerade Educator

Problem 3

For each of these relations on the set $\{1,2,3,4\},$ decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive.
$$
\begin{array}{l}{\text { a) }\{(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)\}} \\ {\text { b) }\{(1,1),(1,2),(2,1),(2,2),(3,3),(3,4)\}} \\ {\text { c) }\{(2,4),(4,2)\}} \\ {\text { d) }\{(1,2),(2,3),(3,4)\}} \\ {\text { e) }\{(1,1),(2,2),(3,3),(4,4)\}} \\ {\text { f) }\{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)\}}\end{array}
$$

James C.
Numerade Educator

Problem 4

Determine whether the relation $R$ on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where $(a, b) \in R$ if and only if
a) $a$ is taller than $b$.
b) $a$ and $b$ were born on the same day.
c) $a$ has the same first name as $b$ .
d) $a$ and $b$ have a common grandparent.

Mj S.
Numerade Educator

Problem 5

Determine whether the relation $R$ on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where $(a, b) \in R$ if and only if
a) everyone who has visited Web page $a$ has also visited Web page $b$ .
b) there are no common links found on both Web page $a$ and Web page $b$ .
c) there is at least one common link on Web page $a$ and Web page $b .$
d) there is a Web page that includes links to both Web page $a$ and Web page $b$ .

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Problem 6

Determine whether the relation $R$ on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where $(x, y) \in R$ if and only if
a) $x+y=0$
b) $x=\pm y$
c) $x-y$ is a rational number
d) $x=2 y$
e) $x y \geq 0$
f) $x y=0$
g) $x=1$
h) $x=1$ or $y=1$

Mj S.
Numerade Educator

Problem 7

Determine whether the relation $R$ on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where $(x, y) \in R$ if and only if
a) $x \neq y$
b) $x y \geq 1$
c) $x=y+1$ or $x=y-1$
d) $x \equiv y(\bmod 7)$
e) $x$ is a multiple of $y$
f) $x$ and $y$ are both negative or both nonnegative.
g) $x=y^{2}$
h) $x \geq y^{2}$

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Problem 8

Show that the relation $R=\emptyset$ on a nonempty set $S$ is symmetric and transitive, but not reflexive.

Mj S.
Numerade Educator

Problem 9

Show that the relation $R=\emptyset$ on the empty set $S=\emptyset$ is reflexive, symmetric, and transitive.

James C.
Numerade Educator

Problem 10

Give an example of a relation on a set that is
a) both symmetric and antisymmetric.
b) neither symmetric nor antisymmetric.

Mj S.
Numerade Educator

Problem 11

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Which relations in Exercise 3 are irreflexive?

James C.
Numerade Educator

Problem 12

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Which relations in Exercise 4 are irreflexive?

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Problem 13

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Which relations in Exercise 5 are irreflexive?

James C.
Numerade Educator

Problem 14

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Which relations in Exercise 6 are irreflexive?

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Problem 15

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Can a relation on a set be neither reflexive nor irreflexive?

James C.
Numerade Educator

Problem 16

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Use quantifiers to express what it means for a relation to be irreflexive.

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Problem 17

A relation $R$ on the set $A$ is irreflexive if for every $a \in A,(a, a) \notin R .$ That is, $R$ is irreflexive if no element in $A$ is related to itself.
Give an example of an irreflexive relation on the set of all people.

James C.
Numerade Educator

Problem 18

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Which relations in Exercise 3 are asymmetric?

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Problem 19

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Which relations in Exercise 4 are asymmetric?

James C.
Numerade Educator

Problem 20

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Which relations in Exercise 5 are asymmetric?

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Problem 21

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Which relations in Exercise 6 are asymmetric?

James C.
Numerade Educator

Problem 22

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric? Give reasons for your answers.

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Problem 23

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Use quantifiers to express what it means for a relation to be asymmetric.

James C.
Numerade Educator

Problem 24

A relation $R$ is called asymmetric if $(a, b) \in R$ implies that $(b, a) \notin R .$ Exercises $18-24$ explore the notion of an asymmetric relation. Exercise 22 focuses on the difference between asymmetry and antisymmetry.
Give an example of an asymmetric relation on the set of all people.

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Problem 25

How many different relations are there from a set with $m$ elements to a set with $n$ elements?

James C.
Numerade Educator

Problem 26

Let $R$ be a relation from a set $A$ to a set $B$ . The inverse relation from $B$ to $A,$ denoted by $R^{-1}$ , is the set of ordered pairs $\{(b, a) |(a, b) \in R\} .$ The complementary relation $\overline{R}$ is the set of ordered pairs $\{(a, b) |(a, b) \notin R\}$.
Let $R$ be the relation $R=\{(a, b) | a<b\}$ on the set of integers. Find
$\begin{array}{ll}{\text { a) } R^{-1}} & {\text { b) } \overline{R}}\end{array}$

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Problem 27

Let $R$ be a relation from a set $A$ to a set $B$ . The inverse relation from $B$ to $A,$ denoted by $R^{-1}$ , is the set of ordered pairs $\{(b, a) |(a, b) \in R\} .$ The complementary relation $\overline{R}$ is the set of ordered pairs $\{(a, b) |(a, b) \notin R\}$.
Let $R$ be the relation $R=\{(a, b) | a \text { divides } b\}$ on the set of positive integers. Find
$\begin{array}{ll}{\text { a) } R^{-1} .} & {\text { b) } \overline{R}}\end{array}$

James C.
Numerade Educator

Problem 28

Let $R$ be a relation from a set $A$ to a set $B$ . The inverse relation from $B$ to $A,$ denoted by $R^{-1}$ , is the set of ordered pairs $\{(b, a) |(a, b) \in R\} .$ The complementary relation $\overline{R}$ is the set of ordered pairs $\{(a, b) |(a, b) \notin R\}$.
Let $R$ be the relation on the set of all states in the United States consisting of pairs $(a, b)$ where state $a$ borders state $b .$ Find
$\begin{array}{ll}{\text { a) } R^{-1}} & {\text { b) } \overline{R}}\end{array}$

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Problem 29

Suppose that the function $f$ from $A$ to $B$ is a one-to-one correspondence. Let $R$ be the relation that equals the graph of $f .$ That is, $R=\{(a, f(a)) | a \in A\} .$ What is the inverse relation $R^{-1} ?$

James C.
Numerade Educator

Problem 30

Let $R_{1}=\{(1,2),(2,3),(3,4)\}$ and $R_{2}=\{(1,1),(1,2)$ $(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4) \}$ be relations from $\{1,2,3\}$ to $\{1,2,3,4\} .$ Find
$$
\begin{array}{ll}{\text { a) } R_{1} \cup R_{2}} & {\text { b) } R_{1} \cap R_{2}} \\ {\text { c) } R_{1}-R_{2}} & {\text { d) } R_{2}-R_{1}}\end{array}
$$

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Problem 31

Let $A$ be the set of students at your school and $B$ the set of books in the school library. Let $R_{1}$ and $R_{2}$ be the relations consisting of all ordered pairs $(a, b),$ where student $a$ is required to read book $b$ in a course, and where student $a$ is required to read book $b$ in a course, and where student $a$ has read book $b$ , respectively. Describe the ordered pairs in each of these relations.
$$
\begin{array}{ll}{\text { a) } R_{1} \cup R_{2}} & {\text { b) } R_{1} \cap R_{2}} \\ {\text { c) } R_{1} \oplus R_{2}} & {\text { d) } R_{1}-R_{2}} \\ {\text { e) } R_{2}-R_{1}}\end{array}
$$

James C.
Numerade Educator

Problem 32

Let $R$ be the relation $\{(1,2),(1,3),(2,3),(2,4),(3,1)\}$ and let $S$ be the relation $\{(2,1),(3,1),(3,2),(4,2)\} .$ Find $S \circ R .$

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Problem 33

Let $R$ be the relation on the set of people consisting of pairs $(a, b),$ where $a$ is a parent of $b$ . Let $S$ be the relation on the set of people consisting of pairs $(a, b),$ where $a$ and $b$ are siblings (brothers or sisters). What are $S \circ R$ and $R \circ S ?$

James C.
Numerade Educator

Problem 34

Exercises $34-38$ deal with these relations on the set of real numbers:
$\begin{aligned} R_{1}=&\left\{(a, b) \in \mathbf{R}^{2} | a>b\right\}, \text { the greater than relation, } \\ R_{2}=&\left\{(a, b) \in \mathbf{R}^{2} | a \geq b\right\}, \text { the greater than or equal to relation, } \end{aligned}$
$\begin{aligned} R_{3}=\left\{(a, b) \in \mathbf{R}^{2} | a < b\right\}, \text { the less than relation, } \\ R_{4}= \left\{(a, b) \in \mathbf{R}^{2} | a \leq b\right\}, \text { the less than or equal to relation, } \end{aligned}$
$R_{5}=\left\{(a, b) \in \mathbf{R}^{2} | a=b\right\},$ the equal to relation,
$R_{6}=\left\{(a, b) \in \mathbf{R}^{2} | a \neq b\right\},$ the unequal to relation.
Find
$$
\begin{array}{lll}{\text { a) } R_{1} \cup R_{3}} & {\text { b) } R_{1} \cup R_{5}} \\ {\text { c) } R_{2} \cap R_{4}} & {\text { d) } R_{3} \cap R_{5}} \\ {\text { e) } R_{1}-R_{2}} & {\text { f) } R_{2}-R_{1}} \\ {\text { g) } R_{1} \oplus R_{3}} & {\text { h) } R_{2} \oplus R_{4}}\end{array}
$$

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Problem 35

Exercises $34-38$ deal with these relations on the set of real numbers:
$\begin{aligned} R_{1}=&\left\{(a, b) \in \mathbf{R}^{2} | a>b\right\}, \text { the greater than relation, } \\ R_{2}=&\left\{(a, b) \in \mathbf{R}^{2} | a \geq b\right\}, \text { the greater than or equal to relation, } \end{aligned}$
$\begin{aligned} R_{3}=\left\{(a, b) \in \mathbf{R}^{2} | a < b\right\}, \text { the less than relation, } \\ R_{4}= \left\{(a, b) \in \mathbf{R}^{2} | a \leq b\right\}, \text { the less than or equal to relation, } \end{aligned}$
$R_{5}=\left\{(a, b) \in \mathbf{R}^{2} | a=b\right\},$ the equal to relation,
$R_{6}=\left\{(a, b) \in \mathbf{R}^{2} | a \neq b\right\},$ the unequal to relation.
Find
$$
\begin{array}{lll}{\text { a) } R_{2} \cup R_{4}} & {\text { b) } R_{3} \cup R_{6}} \\ {\text { c) } R_{3} \cap R_{6}} & {\text { d) } R_{4} \cap R_{6}} \\ {\text { e) } R_{3}-R_{6}} & {\text { f) } R_{6}-R_{3}} \\ {\text { g) } R_{2} \oplus R_{6}} & {\text { h) } R_{3} \oplus R_{5}}\end{array}
$$

James C.
Numerade Educator

Problem 36

Exercises $34-38$ deal with these relations on the set of real numbers:
$\begin{aligned} R_{1}=&\left\{(a, b) \in \mathbf{R}^{2} | a>b\right\}, \text { the greater than relation, } \\ R_{2}=&\left\{(a, b) \in \mathbf{R}^{2} | a \geq b\right\}, \text { the greater than or equal to relation, } \end{aligned}$
$\begin{aligned} R_{3}=\left\{(a, b) \in \mathbf{R}^{2} | a < b\right\}, \text { the less than relation, } \\ R_{4}= \left\{(a, b) \in \mathbf{R}^{2} | a \leq b\right\}, \text { the less than or equal to relation, } \end{aligned}$
$R_{5}=\left\{(a, b) \in \mathbf{R}^{2} | a=b\right\},$ the equal to relation,
$R_{6}=\left\{(a, b) \in \mathbf{R}^{2} | a \neq b\right\},$ the unequal to relation.
Find
$$
\begin{array}{ll}{\text { a) } R_{1} \circ R_{1} .} & {\text { b) } R_{1} \circ R_{2}} \\ {\text { c) } R_{1} \circ R_{3} .} & {\text { d) } R_{1} \circ R_{4}} \\ {\text { e) } R_{1} \circ R_{5} .} & {\text { f) } R_{1} \circ R_{6}} \\ {\text { g) } R_{2} \circ R_{3} .} & {\text { h) } R_{3} \circ R_{3}}\end{array}
$$

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Problem 37

Exercises $34-38$ deal with these relations on the set of real numbers:
$\begin{aligned} R_{1}=&\left\{(a, b) \in \mathbf{R}^{2} | a>b\right\}, \text { the greater than relation, } \\ R_{2}=&\left\{(a, b) \in \mathbf{R}^{2} | a \geq b\right\}, \text { the greater than or equal to relation, } \end{aligned}$
$\begin{aligned} R_{3}=\left\{(a, b) \in \mathbf{R}^{2} | a < b\right\}, \text { the less than relation, } \\ R_{4}= \left\{(a, b) \in \mathbf{R}^{2} | a \leq b\right\}, \text { the less than or equal to relation, } \end{aligned}$
$R_{5}=\left\{(a, b) \in \mathbf{R}^{2} | a=b\right\},$ the equal to relation,
$R_{6}=\left\{(a, b) \in \mathbf{R}^{2} | a \neq b\right\},$ the unequal to relation.
Find
$$
\begin{array}{llll}{\text { a) } R_{2} \circ R_{1}} & {\text { b) } R_{2} \circ R_{2}} \\ {\text { c) } R_{3} \circ R_{5}} & {\text { d) } R_{4} \circ R_{1}} \\ {\text { e) } R_{5} \circ R_{3}} & {\text { f) } R_{3} \circ R_{6}} \\ {\text { g) } R_{4} \circ R_{6}} & {\text { h) } R_{6} \circ R_{6}}\end{array}
$$

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Problem 38

Exercises $34-38$ deal with these relations on the set of real numbers:
$\begin{aligned} R_{1}=&\left\{(a, b) \in \mathbf{R}^{2} | a>b\right\}, \text { the greater than relation, } \\ R_{2}=&\left\{(a, b) \in \mathbf{R}^{2} | a \geq b\right\}, \text { the greater than or equal to relation, } \end{aligned}$
$\begin{aligned} R_{3}=\left\{(a, b) \in \mathbf{R}^{2} | a < b\right\}, \text { the less than relation, } \\ R_{4}= \left\{(a, b) \in \mathbf{R}^{2} | a \leq b\right\}, \text { the less than or equal to relation, } \end{aligned}$
$R_{5}=\left\{(a, b) \in \mathbf{R}^{2} | a=b\right\},$ the equal to relation,
$R_{6}=\left\{(a, b) \in \mathbf{R}^{2} | a \neq b\right\},$ the unequal to relation.
Find the relations $R_{i}^{2}$ for $i=1,2,3,4,5,6$

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Problem 39

Find the relations $S_{i}^{2}$ for $i=1,2,3,4,5,6$ where
$$
\begin{aligned} S_{1}=&\left\{(a, b) \in \mathbf{Z}^{2} | a>b\right\}, \text { the greater than relation, } \\ S_{2}=&\left\{(a, b) \in \mathbf{Z}^{2} | a \geq b\right\}, \text { the greater than or equal to } \\ & \text { relation, } \end{aligned}
$$
$$
\begin{aligned} S_{3}=&\left\{(a, b) \in \mathbf{Z}^{2} | a<b\right\}, \text { the less than relation, } \\ S_{4}=&\left\{(a, b) \in \mathbf{Z}^{2} | a \leq b\right\}, \text { the less than or equal to } \\ & \text { relation, } \end{aligned}
$$
$$
\begin{array}{l}{S_{5}=\left\{(a, b) \in \mathbf{Z}^{2} | a=b\right\}, \text { the equal to relation, }} \\ {S_{6}=\left\{(a, b) \in \mathbf{Z}^{2} | a \neq b\right\}, \text { the unequal to relation. }}\end{array}
$$

James C.
Numerade Educator

Problem 40

Let $R$ be the parent relation on the set of all people (see Example 21 ). When is an ordered pair in the relation $R^{3} ?$

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Problem 41

Let $R$ be the relation on the set of people with doctorates such that $(a, b) \in R$ if and only if $a$ was the thesis advisor of $b .$ When is an ordered pair $(a, b)$ in $R^{2} ?$ When is an ordered pair $(a, b)$ in $R^{n},$ when $n$ is a positive integer? (Assume that every person with a doctorate has a thesis advisor.)

James C.
Numerade Educator

Problem 42

Let $R_{1}$ and $R_{2}$ be the "divides" and "is a multiple of relations on the set of all positive integers, respectively. That is, $R_{1}=\{(a, b) | a \text { divides } b\}$ and $R_{2}=\{(a, b) | a$ is a multiple of $b \}$ . Find
$$
\begin{array}{ll}{\text { a) } R_{1} \cup R_{2} .} & {\text { b) } R_{1} \cap R_{2}} \\ {\text { c) } R_{1}-R_{2}} & {\text { d) } R_{2}-R_{1}} \\ {\text { e) } R_{1} \oplus R_{2}}\end{array}
$$

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Problem 43

Let $R_{1}$ and $R_{2}$ be the "congruent modulo 3 " and the "congruent modulo 4 " relations, respectively, on the set of integers. That is, $R_{1}=\{(a, b) | a \equiv b(\bmod 3)\}$ and $R_{2}=$
$\{(a, b) | a \equiv b(\bmod 4)\} .$ Find
$$
\begin{array}{ll}{\text { a) } R_{1} \cup R_{2} .} & {\text { b) } R_{1} \cap R_{2}} \\ {\text { c) } R_{1}-R_{2} .} & {\text { d) } R_{2}-R_{1}} \\ {\text { e) } R_{1} \oplus R_{2}}\end{array}
$$

James C.
Numerade Educator

Problem 44

List the 16 different relations on the set $\{0,1\}$.

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Problem 45

How many of the 16 different relations on $\{0,1\}$ contain the pair $(0,1) ?$

James C.
Numerade Educator

Problem 46

Which of the 16 relations on $\{0,1\},$ which you listed in Exercise $44,$ are
$$
\begin{array}{ll}{\text { a) reflexive? }} & {\text { b) irreflexive? }} \\ {\text { c) symmetric? }} & {\text { d) antisymmetric? }} \\ {\text { e) asymmetric? }} & {\text { f) transitive? }}\end{array}
$$

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Problem 47

a) How many relations are there on the set $\{a, b, c, d\} ?$
b) How many relations are there on the set $\{a, b, c, d\}$ that contain the pair $(a, a) ?$

James C.
Numerade Educator

Problem 48

Let $S$ be a set with $n$ elements and let $a$ and $b$ be distinct elements of $S .$ How many relations $R$ are there on $S$ such that
a) $(a, b) \in R ? \quad$ b) $(a, b) \notin R ?$
c) no ordered pair in $R$ has $a$ as its first element?
d) at least one ordered pair in $R$ has $a$ as its first element?
e) no ordered pair in $R$ has $a$ as its first element or $b$ as its second element?
f) at least one ordered pair in $R$ either has $a$ as its first element or has $b$ as its second element?

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Problem 49

How many relations are there on a set with $n$ elements that are
$\begin{array}{ll}{\text { a) symmetric? }} & {\text { b) antisymmetric? }} \\ {\text { c) asymmetric? }} & {\text { d) irreflexive? }}\end{array}$
e) reflexive and symmetric?
f) neither reflexive nor irreflexive?

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Problem 50

How many transitive relations are there on a set with $n$ elements if
$\begin{array}{llll}{\text { a) } n=1 ?} & {\text { b) } n=2 ?} & {\text { c) } n=3 ?}\end{array}$

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Problem 51

Find the error in the "proof' of the following "theorem." "Theorem": Let $R$ be a relation on at $A$ that is symmetric and transitive. Then $R$ is reflexive. "Proof": Let $a \in A$ . Take an element $b \in A$ such that $(a, b) \in R .$ Because $R$ is symmetric, we also have $(b, a) \in$ $R .$ Now using the transitive property, we can conclude that $(a, a) \in R$ because $(a, b) \in R$ and $(b, a) \in R .$

James C.
Numerade Educator

Problem 52

Suppose that $R$ and $S$ are reflexive relations on a set $A .$ Prove or disprove each of these statements.
a) $R \cup S$ is reflexive.
b) $R \cap S$ is reflexive.
c) $R \oplus S$ is irreflexive.
d) $R-S$ is irreflexive.
e) $S \circ R$ is reflexive.

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Problem 53

Show that the relation $R$ on a set $A$ is symmetric if and only if $R=R^{-1},$ where $R^{-1}$ is the inverse relation.

James C.
Numerade Educator

Problem 54

Show that the relation $R$ on a set $A$ is antisymmetric if and only if $R \cap R^{-1}$ is a subset of the diagonal relation $\Delta=\{(a, a) | a \in A\}$.

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Problem 55

Show that the relation $R$ on a set $A$ is reflexive if and only if the inverse relation $R^{-1}$ is reflexive.

James C.
Numerade Educator

Problem 56

Show that the relation $R$ on a set $A$ is reflexive if and only if the complementary relation $\overline{R}$ is irreflexive.

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Problem 57

Let $R$ be a relation that is reflexive and transitive. Prove that $R^{n}=R$ for all positive integers $n .$

James C.
Numerade Educator

Problem 58

Let $R$ be the relation on the set $\{1,2,3,4,5\}$ containing the ordered pairs $(1,1),(1,2),(1,3),(2,3),(2,4),(3,1),$ $(3,4),(3,5),(4,2),(4,5),(5,1),(5,2),$ and $(5,4) .$ Find
$\begin{array}{llll}{\text { a) } R^{2}} & {\text { b) } R^{3} .} & {\text { c) } R^{4}} & {\text { d) } R^{5}}\end{array}$

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Problem 59

Let $R$ be a reflexive relation on a set $A .$ Show that $R^{n}$ is reflexive for all positive integers $n .$

James C.
Numerade Educator

Problem 60

Let $R$ be a symmetric relation. Show that $R^{n}$ is symmetric for all positive integers $n .$

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Problem 61

Suppose that the relation $R$ is irreflexive. Is $R^{2}$ necessarily irreflexive? Give a reason for your answer.

James C.
Numerade Educator

Problem 62

Derive a big- $O$ estimate for the number of integer comparisons needed to count all transitive relations on a set with $n$ elements using the brute force approach of checking every relation of this set for transitivity.

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