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

Suggested languages for you:

Americas

Europe

Q21E

Expert-verified
Found in: Page 582

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Determine whether the relation R on the set of all real numbers are asymmetric.

The given relation is not asymmetric.

See the step by step solution

## Step 1: Given data

$x+y=0$

## 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 ${a}{,}{b}{\in }{A}$, 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

An asymmetric relation is one for which $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ can never hold simultaneously, even if a=b.

Thus $R$ is asymmetric if and only if R is anti symmetric and also irreflexive.

$\left(x,y\right)\in R$

$x+y=0$

$y+x=0$

$\left(y,x\right)\in R$

Thus, R is not asymmetric.