Question

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

Short answer

$\forall a\forall b\left(\right(a,b)\in R\to (b,a)\notin R)$

Step-by-step solution

Given data

An asymmetric relation.

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

We know that $\text{R}$ is called asymmetric if $(a,b)\in \text{R}$ implies that $(b,a)\notin \text{R}$.

Using quantifiers we see that a relation $R$ on the set $A$is asymmetric if we have $\forall a\forall b\left(\right(a,b)\in R\to (b,a)\notin R)$ where, set of all elements $A$.

Quantifiers to express what it means for a relation to be asymmetric is $\forall a\forall b\left(\right(a,b)\in R\to (b,a)\notin R)$.