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Expert-verified Found in: Page 582 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # Find $\overline{\mathbf{R}}$ if${\mathbit{R}}{\mathbf{=}}{\mathbf{\left\{}}{\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\right)}}{\mathbf{\mid }}{\mathbit{a}}{\mathbf{<}}{\mathbit{b}}{\mathbf{\right\}}}$.

$\overline{R}=\left\{\left(a,b\right)\mid a\ge b\right\}$

See the step by step solution

## Step 1: Given data

The relation $R$ is $R=\left\{\left(a,b\right)\mid a.

## Step 2: Concept used of relation

A relation ${\mathbf{\text{R}}}$ on a set ${\mathbf{\text{A}}}$ is called reflexive if ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{a}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$for every element ${\mathbit{a}}{\mathbf{\in }}{\mathbit{A}}$.

A relation role="math" localid="1668686721462" ${\mathbf{\text{R}}}$ on a set ${\mathbf{\text{A}}}$ is called symmetric if ${\mathbf{\left(}}{\mathbit{b}}{\mathbf{,}}{\mathbit{a}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ whenever ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$, for all ${\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\in }}{\mathbit{A}}$

A relation ${\mathbf{\text{R}}}$ on a set ${\mathbf{\text{A}}}$ such that for all ${\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\in }}{\mathbit{A}}$, if ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ and ${\mathbf{\left(}}{\mathbit{b}}{\mathbf{,}}{\mathbit{a}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ then ${\mathbit{a}}{\mathbf{=}}{\mathbit{b}}$ is called anti symmetric.

A relation ${\mathbf{\text{R}}}$ on a set ${\mathbf{\text{A}}}$ is called transitive if whenever ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ and ${\mathbf{\left(}}{\mathbit{b}}{\mathbf{,}}{\mathbit{c}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ then ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{c}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$for all ${\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{,}}{\mathbit{c}}{\mathbf{\in }}{\mathbit{A}}$

## Step 3: Solve for relation

The relation $R=\left\{\left(a,b\right)\mid \left(a 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 $\left(a,b\right)\notin R$. An ordered pair of the form $\left(a,b\right)$ will belong to relation $R$ if the condition $a holds that is If $a, then $\left(a,b\right)\in R$.

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

Thus,

$R=\left\{\left(a,b\right)\mid \left(a

$\overline{R}=\left\{\left(a,b\right)\mid a\ge b\right\}$

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 $\overline{R}=\left\{\left(a,b\right)\mid a\ge b\right\}$. ### Want to see more solutions like these? 