Question

Show that the relation$\mathit{R}\mathbf{=}\mathit{\varphi }$on a non-empty set $\mathbf{\text{S}}$is symmetric and transitive, but not reflexive.

Short answer

Hence$\text{R}$is symmetric, transitive and not reflexive.

Step-by-step solution

Given data

The relation $R=\varphi$on a non-empty set 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 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.

Not reflexive For any element $a\in S:(a,a)\notin R$because $\text{R}$is the empty set. By the definition of reflexive, $R$is then not reflexive.

Hence $\text{R}$is symmetric, transitive and not reflexive.