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Q18E

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Discrete Mathematics and its Applications

Found in: Page 582

Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Which relations in Exercise 3 are asymmetric?

The sets which are asymmetric from Exercise $3$ are below.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

${1, 2, 3, 4}$ is the given set.

Step 2: Concept used of relation

A relation $R$ on a set $A$ is called reflexive if $(a, a) \in R$ for every element $a \in A$ .

A relation $R$ on a set $A$ is called symmetric if $(b, a) \in R$ whenever $(a, b) \in R$ , for all $a, b \in A$

A relation $R$ on a set $A$ such that for all $a, b \in A$ , if $(a, b) \in R$ and $(b, a) \in R$ then $a = b$ is called anti symmetric.

A relation $R$ on a set $A$ is called transitive if whenever $(a, b) \in R$ and $(b, c) \in R$ then $(a, c) \in R$ for all $a, b, c \in A$

Step 3: Solve for relation

A relation is asymmetric if and only if it is both antisymmetric and irreflexive. ${(1, 2), (2, 3), (3, 4)}$ is the only set which is asymmetric.

Most popular questions for Math Textbooks

To determine an example of an asymmetric relation on the set of all people.

Whether there is a path in the directed graph in Exercise 16 beginning at the first vertex given and ending at the second vertex given.

Can a relation on a set be neither reflexive nor irreflexive?

Use quantifiers to express what it means for a relation to be asymmetric.

Exercises 34–37 deal with these relations on the set of real numbers:

\({R_1} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a > b} \right\},\) the “greater than” relation,

\({R_2} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ge b} \right\},\) the “greater than or equal to” relation,

\({R_3} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a < b} \right\},\) the “less than” relation,

\({R_4} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \le b} \right\},\) the “less than or equal to” relation,

\({R_5} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a = b} \right\},\) the “equal to” relation,

\({R_6} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ne b} \right\},\) the “unequal to” relation.

36. Find

(a) \({R_1}^\circ {R_1}\).

(b) \({R_1}^\circ {R_2}\).

(c) \({R_1}^\circ {R_3}\).

(d) \({R_1}^\circ {R_4}\).

(e) \({R_1}^\circ {R_5}\).

(f) \({R_1}^\circ {R_6}\).

(g) \({R_2}^\circ {R_3}\).

(h) \({R_3}^\circ {R_3}\).

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