Question

Which relations in Exercise 3 are asymmetric?

Short answer

The sets which are asymmetric from Exercise $3$are below.

Step-by-step solution

Given data

$\{1,2,3,4\}$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

A relation is asymmetric if and only if it is both antisymmetric and irreflexive. $\left\{\right(1,2),(2,3),(3,4\left)\right\}$is the only set which is asymmetric.