Question

Which relations in Exercise $\mathbf{6}$are irreflexive?

Short answer

The sets which are irreflexive from Exercise$6$ are below.

Step-by-step solution

Given data

$(x,y)\in R$ 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 role="math" localid="1668689181702" $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{A}$.

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

$x+y=0$

$x=2y$

$xy=0$

$x=1$

$x=1\text{or}y=1$

These are the sets which are irreflexive.