Question

Determine whether the relation R on the set of all real numbers are asymmetric.

Short answer

The given relation is not asymmetric.

Step-by-step solution

Given data

$x+y=0$

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 $a,b\in A$, 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 a=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

An asymmetric relation is one for which $(a,b)\in R$and $(b,a)\in R$can never hold simultaneously, even if a=b.

Thus $R$is asymmetric if and only if R is anti symmetric and also irreflexive.

$(x,y)\in R$

$x+y=0$

$y+x=0$

$(y,x)\in R$

Thus, R is not asymmetric.