Chapter 3.5, Problem 1E

(a)

To determine

To find whether therelation R is antisymmetric and has comparability property.

Answer to Problem 1E

The given relation Ris neither antisymmetric nor has comparability property.

Explanation of Solution

Given information: The relation R defined on the set $A=\left\{1,2,3,4,5\right\}$ is $R=\left\{\left(1,3\right),\left(1,1\right),\left(2,4\right),\left(3,2\right),\left(5,4\right),\left(4,2\right),\right\}$

  $A=\left\{1,2,3,4,5\right\}$ .

Definition used:

• A relation R on a set A is said to have comparability property then, every two elements on Ashould be comparable.
• A relation R on a set Ais antisymmetric if and only if for all $x,y\in A,\text{if }xRy\text{ and }xRy,\text{ then}x=y$

Calculation:

The given relation is not antisymmetric since, $\left(2,4\right)\text{ and }\left(4,2\right)$ belongs to R but $2\ne 4$

The given relation R doesn’t have comparability property since $\left(1,2\right),\left(2,5\right),\left(1,5\right),\left(5,3\right)$ doesn’t belong to R.

b.

To determine

To find whether the relation R is antisymmetric and has comparability property.

Answer to Problem 1E

The given relation R is antisymmetric and doesn’t have comparability property

Explanation of Solution

Given information: The relation R defined on the set $A=\left\{1,2,3,4,5\right\}$ is $\left\{\left(1,4\right),\left(1,2\right),\left(2,3\right),\left(3,4\right),\left(5,2\right),\left(4,2\right),\left(1,3\right)\right\}$ .

Definition used:

• A relation R on a set A is said to have comparability property then, every two elements on Ashould be comparable.
• A relation R on a set A is antisymmetric if and only if for all $x,y\in A,\text{if }xRy\text{ and }xRy,\text{ then}x=y$

Calculation:

The given relation is antisymmetric since, for every $x,y\in A,xRy$ but there doesnot

exist $yRx$ .

The given relation doesn’t have comparability property since $\left(1,5\right),\left(3,5\right),\left(4,5\right)$ doesn’t belong to the relation.

c.

To determine

To find whether the relation R is antisymmetric and has comparability property.

Answer to Problem 1E

The given relation R is not antisymmetric and doesn’t have comparability property

Explanation of Solution

Given information: The relation R defined on the set $\mathbb{Z}$ is $xRy{\text{ if x}}^2=y^2$ .

Definition used:

• A relation R on a set A is said to have comparability property then, every two elements on Ashould be comparable.
• A relation R on a set A is antisymmetric if and only if for all $x,y\in A,\text{if }xRy\text{ and }xRy,\text{ then}x=y$

Calculation:

The given relation is not antisymmetric.

Example: Let $x=3$ and $y=-3$

  $\begin{array}{c}{\left(3\right)}^2=9={\left(-x\right)}^2,xRy\\ \text{and }{\left(-3\right)}^2=9={\left(3\right)}^2,yRx\end{array}$

But $x\ne y$

The given relation doesn’t have comparability property.

Example: $1,2\in A{\text{ but 1}}^2\ne 2^2$ . Hence $\left(1,2\right)\notin R$ .

d.

To determine

To find whether the relation R is antisymmetric and has comparability property.

Answer to Problem 1E

The given relation R is not antisymmetric and has comparability property

Explanation of Solution

Given information: The relation R defined on the set $ℝ\text{ is }xRy\text{ if }x\le 2^y$ .

Definition used:

• A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.
• A relation R on a set A is antisymmetric if and only if for all $x,y\in A,\text{if }xRy\text{ and }xRy,\text{ then}x=y$

Calculation:

The given relation is not antisymmetric.

Example: Let $x=3\text{ and }y=4$

  $\begin{array}{l}3\le 2^4,\text{ so }xRy\\ 4\le 2^3,\text{ so }yRx\\ \text{But 3}\ne \text{4 }\end{array}$

The given relation has comparability property because for all $x,y\in ℝ\text{ either }xRy\text{or }yRx$ .

e.

To determine

To find whether the relation R is antisymmetric and has comparability property.

Answer to Problem 1E

The given relation R is antisymmetric and doesn’t have comparability property.

Explanation of Solution

Given information: The relation R defined on the set $ℝ\times ℝ$ is $xSy\text{if }y=x-1$ .

Definition used:

• A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.
• A relation R on a set A is antisymmetric if and only if for all $x,y\in A,\text{if }xRy\text{ and }xRy,\text{ then}x=y$

Calculation:

The given relation is antisymmetric.

For $x,y\in ℝ\times ℝ$ , $y=x-1\text{and }x=y-1$ cannot happen simultaneously.

In other words, if $y=x-1\text{and }x=y-1$ , then $x=y$ is true for all $x,y\in ℝ\times ℝ$ .

The given relation doesn’t have comparability property since there are indefinite numbers in $ℝ\times ℝ$ such that $y=x-1\text{and }x=y-1$ is false.

f.

To determine

To find whether the relation R is antisymmetric and has comparability property.

Answer to Problem 1E

The given relation R is not antisymmetric and doesn’t have comparability property

Explanation of Solution

Given information: The relation R defined on the set $A=\left\{1,2,3,4\right\}$ is shown in the digraph below:

  

Definition used:

• A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.
• A relation R on a set A is antisymmetric if and only if for all $x,y\in A,\text{if }xRy\text{ and }xRy,\text{ then}x=y$

Calculation:

From the digraph, the relation set can be obtained as:

  $R=\left\{\left(1,1\right),\left(1,4\right),\left(4,4\right),\left(4,3\right),\left(3,3\right),\left(3,2\right),\left(2,2\right),\left(2,1\right),\left(1,3\right),\left(3,1\right)\right\}$

The given relation is not antisymmetric.

Example:

  $\left(1,3\right),\left(3,1\right)\in R$ , but $1\ne 3$ .

The given relation doesn’t have comparability:

For example $\left(2,4\right)\text{or }\left(4,2\right)$ doesn’t belong to the relation.

g.

To determine

To find whether the relation R is antisymmetric and has comparability property.

Answer to Problem 1E

The given relation R is antisymmetric and has comparability property

Explanation of Solution

The given relation is not antisymmetric sinceGiven information: The relation R defined on the set $A=\left\{1,2,3,4\right\}$ is shown in digraph as:

  

Definition used:

A relation R on a set A is said to have comparability property then, every two elements on A should be comparable.

A relation R on a set A is antisymmetric if and only if for all $x,y\in A,\text{if }xRy\text{ and }xRy,\text{ then}x=y$

Calculation:

From the digraph, the relation set can be obtained as:

  $R=\left\{\left(1,2\right),\left(2,3\right),\left(3,4\right),\left(4,1\right),\left(2,4\right),\left(1,3\right)\right\}$

The given relation is antisymmetric since for every $x,y\in A\text{and}xRy$ but there doesn’t exist $yRx$

The given relation have comparability property since for every $x,y\in A$,either $$