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Found in: Page 582

### 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 asymmetric.

$\forall a\forall b\left(\left(a,b\right)\in R\to \left(b,a\right)\notin R\right)$

See the step by step solution

## Step 1: Given data

An asymmetric relation.

## 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 anti symmetric.

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

We know that $\text{R}$ is called asymmetric if $\left(a,b\right)\in \text{R}$ implies that $\left(b,a\right)\notin \text{R}$.

Using quantifiers we see that a relation $R$ on the set $A$ is asymmetric if we have $\forall a\forall b\left(\left(a,b\right)\in R\to \left(b,a\right)\notin R\right)$ where, set of all elements $A$.

Quantifiers to express what it means for a relation to be asymmetric is $\forall a\forall b\left(\left(a,b\right)\in R\to \left(b,a\right)\notin R\right)$.