Question

To determine whether the relation R on the set of all web pages is reflexive, symmetric, anti symmetric, transitive, where $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$if and only if there is a webpage that includes links to both webpage a and webpage b.

Short answer

The given set is symmetric.

Step-by-step solution

Given data

There is a webpage that includes links to both webpage a and webpage 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$\mathbf{\text{R}}$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 localid="1668673714213" width="70">(a,b)∈Rand width="70">(b,a)∈Rthen a=b is called anti symmetric.

A relation localid="1668673606943" $\mathbf{\text{R}}$on a set width="13">Ais called transitive if whenever localid="1668673619504" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and localid="1668673631595" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$then width="69">(a,c)∈Rfor all width="74">a,b,c∈A

Solve for relation

R would be like: R(a,b)

$\equiv \exists c,a\in$links out of $\left(c\right),b\in$links out of (c)

It is symmetric.