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Q8E

Expert-verifiedFound in: Page 581

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

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

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

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

**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**** ${\mathit{a}}{\mathbf{=}}{\mathit{b}}$ i****s 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****${\mathit{a}}{\mathbf{,}}{\mathit{b}}{\mathbf{,}}{\mathit{c}}{\mathbf{\in}}{\mathit{A}}$**

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.

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