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Q26E

Expert-verifiedFound in: Page 582

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**Find $\overline{\mathbf{R}}$ if${\mathit{R}}{\mathbf{=}}{\mathbf{\left\{}}{\mathbf{\right(}}{\mathit{a}}{\mathbf{,}}{\mathit{b}}{\mathbf{)}}{\mathbf{\mid}}{\mathit{a}}{\mathbf{<}}{\mathit{b}}{\mathbf{\}}}$**.

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

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

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

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

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

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

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

Thus,

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

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

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

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