Question

Show that the relation $\mathit{R}\mathbf{=}\mathit{\varphi }$on a non-empty set $\mathit{S}\mathbf{=}\mathit{\varphi }$is symmetric, transitive and reflexive.

Short answer

Hence $\text{R}$is symmetric, transitive and reflexive.

Step-by-step solution

Given data

The relation_$R=\varphi$on a non-empty set $S=\varphi$is given.

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

Symmetric Since $(a,b)\in R$is always false (as $\text{R}$is the empty set), the conditional statement $\left(\right(a,b)\in R)\to B$is true for any statement $\text{B}$.

Let $B$be the statement " $(b,a)\in R$". The conditional statement

" If $(a,b)\in R$, then $(b,a)\in R$" is then always true and thus the relation $\text{R}$is symmetric by the definition of symmetric.

Transitive Since $(a,b)\in R$is always false (as $\text{R}$is the empty set) and since $(b,a)\in R$is always false, the statement " $(a,b)\in R$, and $(b,a)\in R$" is also always false.

Then the conditional statement $\left(\right(a,b)\in R$and $(b,a)\in R)\to B$is true for any statement B.

Let B the statement " $a=b$". The conditional statement

"If $(a,b)\in R$and $(b,a)\in R$, then $a=b$is then always true and thus the relation $\text{R}$is transitive by the definition of transitive.

Reflexive The definition of a reflexive relation is equivalent with: If $a\in A$, then $(a,a)\in R$.

Since $a\in S$is always false (as $S$is the empty set), the conditional statement $(a\in S)\to B$is true for any statement $B$.

Let B be the statement " $(a,a)\in R$". The conditional statement

" If $a\in S$, then $(a,a)\in R$" is then always true and thus the relation $\text{R}$is reflexive by the definition of reflexive. Hence $R$is symmetric, transitive and reflexive.