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Q18E

Expert-verifiedFound in: Page 582

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

** Which relations in Exercise 3 are asymmetric?**

The sets which are asymmetric from Exercise $3$ are below.

$\{1,2,3,4\}$ is the given set.

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

A relation is asymmetric if and only if it is both antisymmetric and irreflexive. $\left\{\right(1,2),(2,3),(3,4\left)\right\}$ is the only set which is asymmetric.

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