Question

To determine for each of these relations on the set $\mathbf{\{}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{\}}$decide whether it is reflexive, whether it is symmetric, whether it is anti symmetric, and whether it is transitive $\mathbf{\left\{}\mathbf{\right(}\mathbf{1}\mathbf{,}\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{4}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{4}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{\left)}\mathbf{\right\}}$.

Short answer

It is none of the relations.

Step-by-step solution

Given data

$\left\{\right(1,3),(1,4),(2,3),(2,4),(3,1),(3,4\left)\right\}$ 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\in A$.

A relation localid="1668680060143" $\mathbf{\text{R}}$on a set localid="1668680065687" $\mathbf{\text{A}}$is called symmetric if localid="1668680072088" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$whenever localid="1668680076929" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$, for all localid="1668680083313" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$

A relation localid="1668680089366" $\mathbf{\text{R}}$on a set localid="1668680095795" $\mathbf{\text{A}}$such that for all localid="1668680121597" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$, if localid="1668680126839" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and localid="1668680131739" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{a}\mathbf{)}\mathbf{\in }\mathit{R}$then localid="1668680138376" $\mathit{a}\mathbf{=}\mathit{b}$is called anti symmetric.

A relationlocalid="1668680145281" $\mathbf{\text{R}}$on a set localid="1668680150727" $\mathbf{\text{A}}$is called transitive if whenever localid="1668680155927" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{\in }\mathit{R}$and localid="1668680164262" $\mathbf{(}\mathit{b}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$then localid="1668680170361" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$for all localid="1668680175006" $\mathbf{(}\mathit{a}\mathbf{,}\mathit{c}\mathbf{)}\mathbf{\in }\mathit{R}$

Solve for relation

Not reflexive because we do not have (a,a) for all $a=1,2,3,4$.

Not symmetric because the relation does not contain $(4,1),(3,2),(4,2)$, and $(4,3)$.

Not anti symmetric because we have (1,3) and (3,1).

Not transitive because we do not have (2,1) for (2,3) and (3,1).