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Q12E

Expert-verifiedFound in: Page 581

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**Which relations in Exercise 4 are irreflexive?**

The sets which are irreflexive from Exercise 4 are below.

$(a,b)\in R$ 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 ${a}{=}{b}$ 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 ${\mathit{a}}{\mathbf{,}}{\mathit{b}}{\mathbf{,}}{\mathit{c}}{\mathbf{\in}}{\mathit{A}}$**

$a$ is taller than $b$ is the only relation which is irreflexive.

The sets which are irreflexive from Exercise $4$ are above.

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