Question

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

Short answer

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

Step-by-step solution

Given data

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

Concept used of relation

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 $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathit{A}$

Solve for relation

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