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Expert-verified Found in: Page 581 ### Discrete Mathematics and its Applications

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.

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

See the step by step solution

## Step 1: Given data

Consider a relation in a set.

## Step 2: Concept used of relation

A relation${\mathbf{\text{R}}}$on a set ${\mathbf{\text{A}}}$ is called reflexive if ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{a}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ for every element ${\mathbit{a}}{\mathbf{\in }}{\mathbit{A}}$

A relation ${\mathbf{\text{R}}}$ on a set ${\mathbf{\text{A}}}$ is called symmetric if ${\mathbf{\left(}}{\mathbit{b}}{\mathbf{,}}{\mathbit{a}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$whenever ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$, for all ${\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\in }}{\mathbit{A}}$

A relation ${\mathbf{\text{R}}}$ on a set ${\mathbf{\text{A}}}$ such that for all ${\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\in }}{\mathbit{A}}$, if ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$and ${\mathbf{\left(}}{\mathbit{b}}{\mathbf{,}}{\mathbit{a}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$then ${\mathbit{a}}{\mathbf{=}}{\mathbit{b}}$ is called antisymmetric.

A relation ${\mathbf{\text{R}}}$ on a set ${\mathbf{\text{A}}}$is called transitive if whenever ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ and ${\mathbf{\left(}}{\mathbit{b}}{\mathbf{,}}{\mathbit{c}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$then ${\mathbf{\left(}}{\mathbit{a}}{\mathbf{,}}{\mathbit{c}}{\mathbf{\right)}}{\mathbf{\in }}{\mathbit{R}}$ for all ${\mathbit{a}}{\mathbf{,}}{\mathbit{b}}{\mathbf{,}}{\mathbit{c}}{\mathbf{\in }}{\mathbit{A}}$

## Step 3: Solve for relation

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

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

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

$\forall a\in A:\left(a,a\right)\notin R$is the irreflexive relation. ### Want to see more solutions like these? 