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Q16E

Expert-verifiedFound in: Page 581

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**Use quantifiers to express what it means for a relation to be irreflexive.**

$\forall a\in A:(a,a)\notin R$is the irreflexive relation.

Consider a relation in a 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 antisymmetric.**

**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 on a set $\text{A}$ is irreflexive if $(a,a)\notin R$for every element $a\in A$.

For every element can be written mathematically using the quantifier $\forall $

$(a,a)\notin R$for every element $a\in A$ can then be written as:

$\forall a\in A:(a,a)\notin R$is the irreflexive relation.

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