Question

To determine whether the relation $\mathit{R}$ on the set of all real numbers is reflexive, symmetric, anti symmetric, transitive, where $\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{)}\mathbf{\in }\mathit{R}$ if and only ifx=1 or y=1.

Short answer

The given set is symmetric.

Step-by-step solution

Given data

x=1 or y=1 is the given condition and$(x,y)\in R$.

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 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

Not Reflexive $(\neg R(2,2\left)\right)$

Symmetric role="math" localid="1668683716010" $\left(R\right(x,y)$

$R(y,x))$

Not Anti symmetric $\left(R\right(1,2)$and R(2,1) and role="math" localid="1668683610157" $(2\ne 1)$

Not Transitive $\left(R\right(3,1)$and $R(1,2)$ and $\neg R(3,2))$

The given set is symmetric.