Question

Use quantifiers to express what it means for a relation to be irreflexive.

Short answer

$\forall a\in A:(a,a)\notin R$is the irreflexive relation.

Step-by-step solution

Given data

Consider a relation in a 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 antisymmetric.

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 on a set $\text{A}$is irreflexive if $(a,a)\notin R$for every element $a\in A$.

For every element can be written mathematically using the quantifier$\forall$

$(a,a)\notin R$for every element $a\in A$can then be written as:

$\forall a\in A:(a,a)\notin R$is the irreflexive relation.