Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Find $\bar{R}$ if $R = {(a, b) ∣ a < b}$ .

$\bar{R} = {(a, b) ∣ a \geq b}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

The relation $R$ is $R = {(a, b) ∣ a < b}$ .

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 role="math" localid="1668686721462" $R$ 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

The relation $R = {(a, b) ∣ (a < b)}$ is defined on the set of integers. To calculate complementary relation of $R$ there is a need to find all ordered pairs of the form $(a, b) \notin R$ . An ordered pair of the form $(a, b)$ will belong to relation $R$ if the condition $a < b$ holds that is If $a < b$ , then $(a, b) \in R$ .

Then in the complementary relation condition should be complement of $a < b$ . The complement of condition $a < b$ will be $a \geq b$ .

Thus,

$R = {(a, b) ∣ (a < b)}$

$\bar{R} = {(a, b) ∣ a \geq b}$

Thus, the complementary relation contains all such ordered pairs in which first element is greater than or equal to the second.

Hence, the required complementary relation is $\bar{R} = {(a, b) ∣ a \geq b}$ .

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Discrete Mathematics and its Applications

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Short Answer

Find $\bar{R}$ if $R = {(a, b) ∣ a < b}$ .

Step by Step Solution

Step 1: Given data

Step 2: Concept used of relation

Step 3: Solve for relation

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