Question

Which relations in Exercise 5 are irreflexive?

Short answer

None of the set is irreflexive.

Step-by-step solution

Given data

$(a,b)\in R$is the given set.

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 $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$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

An irreflexive relation is one for which a is never related to itself;i.e., $\forall a\left(\right(a,a)\notin R)$.

Everyone who has visited Web page . has also visited Web page ..

Since we saw in Exercise $5a$that $\forall a\left(\right(a,a)\in R)$, clearly$R$is not irreflexive.

There are no common links found on both Web page . and Web page ..

Since there are probably pages a with no links at all, and for such pages it is true that there are no common links found on both page a and page $a$, this relation is probably not irreflexive.

There is at least one common link on Web page . and Web page ..

This relation is not irreflexive, because for any page a that has links on it,$(a,a)\in R$.

There is a Web page that includes links to both Web page . and Web page ..

This relation is not irreflexive, because for any page a that has links on it that are ever cited, $(a,a)\in R$.

None of the set is irreflexive.