• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q12E

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

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.

See the step by step solution

## Step 1: Given data

$\left(a,b\right)\in R$ is the given 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 ${a}{=}{b}$ 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

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

The sets which are irreflexive from Exercise $4$ are above. ### Want to see more solutions like these? 