Question

Find$\overline{\mathbf{R}}$for the given $\mathit{R}$.

Short answer

$R^{-1}=\left\{\right(a,b)\mid a$ does not divide $b\}$.

Step-by-step solution

Given data

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

Concept used of relation

A relation $\mathbf{\text{R}}$ on a set role="math" localid="1668680737364" $\mathbf{\text{A}}$ is called reflexive if role="math" localid="1668680745202" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$ for every element role="math" localid="1668680752031" $\mathit{a}\mathbf{\in }\mathit{A}$.

A relation role="math" localid="1668680762305" $\mathbf{\text{R}}$ on a set role="math" localid="1668680768156" $\mathbf{\text{A}}$ is called symmetric if role="math" localid="1668680775194" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$ whenever role="math" localid="1668680781239" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$, for all role="math" localid="1668680789291" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$

A relation role="math" localid="1668680802212" $\mathbf{\text{R}}$ on a set role="math" localid="1668680818402" $\mathbf{\text{A}}$ such that for all role="math" localid="1668680824707" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$, if role="math" localid="1668680833452" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$ and role="math" localid="1668680851400" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$ then role="math" localid="1668680859896" $\mathit{a}\mathbf{=}\mathit{b}$ is called anti symmetric.

A relation role="math" localid="1668680866580" $\mathbf{\text{R}}$ on a set role="math" localid="1668680873740" $\mathbf{\text{A}}$ is called transitive if whenever role="math" localid="1668680880517" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$ and role="math" localid="1668680887189" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$ then role="math" localid="1668680894116" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$ for all role="math" localid="1668680901128" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathit{A}$

Solve for relation

The complementary relation of $R$is

localid="1668680933655" $$\begin{gathered} \overline{R}=\left\{\right(a,b)/(a,b)\notin R\} \\ \overline{R}=\left\{\right(a,b)/a\text{does not divide}b\} \end{gathered}$$

Hence, the required complementary relation is $\overline{R}=\left\{\right(a,b)\mid a$does not divide $b\}$