Chapter 4.4, Problem 9E

(a)

To determine

Tofind:The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{1,2,3,4,5,6,7\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

For the notation $I_c$ , the permutation of the set is remains same.

Hence, the required permutation set is $\left\{1,2,3,4,5,6,7\right\}$ .

(b)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{5,6,2,4,1,7,3\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

The given notation is $u=\left\{\left(1,5\right),\left(6,7\right),\left(4,4\right),\left(5,1\right),\left(3,2\right),\left(2,6\right),\left(7,3\right)\right\}$ .

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in u$ .

Hence, the required permutation set is $\left\{5,6,2,4,1,7,3\right\}$ .

(c)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{2,5,4,6,1,7,3\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

The given notation is $v=\left\{\left(1,2\right),\left(2,5\right),\left(3,4\right),\left(5,1\right),\left(4,6\right),\left(6,7\right),\left(7,3\right)\right\}$ .

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in v$ .

Hence, the required permutation set is $\left\{2,5,4,6,1,7,3\right\}$ .

(d)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{2,1,4,3,5,7,6\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

The given notation is $w=\left\{\left(1,2\right),\left(2,1\right),\left(3,4\right),\left(4,3\right),\left(5,5\right),\left(6,7\right),\left(7,6\right)\right\}$ .

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in w$ .

Hence, the required permutation set is $\left\{2,1,4,3,5,7,6\right\}$ .

(e)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{6,1,4,7,5,3,2\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $u\circ v$ .

  $u\circ v=\left\{\left(1,1\right),\left(2,7\right),\left(3,5\right),\left(4,6\right),\left(5,2\right),\left(6,3\right),\left(7,4\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in u\circ v$ .

Hence, the required permutation set is $\left\{6,1,4,7,5,3,2\right\}$ .

(f)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{1,7,5,6,2,3,4\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $v\circ u$ .

  $v\circ u=\left\{\left(1,1\right),\left(2,7\right),\left(3,5\right),\left(4,6\right),\left(5,2\right),\left(6,3\right),\left(7,4\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in v\circ u$ .

Hence, the required permutation set is $\left\{1,7,5,6,2,3,4\right\}$ .

(g)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{1,2,3,4,5,6,7\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $w\circ w$ .

  $w\circ w=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right),\left(5,5\right),\left(6,6\right),\left(7,7\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in w\circ w$ .

Hence, the required permutation set is $\left\{1,2,3,4,5,6,7\right\}$ .

(h)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{1,6,7,4,5,2,3\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $u\circ v\circ w$ .

  $u\circ v\circ w=\left\{\left(1,1\right),\left(2,6\right),\left(3,7\right),\left(4,4\right),\left(5,5\right),\left(6,2\right),\left(7,3\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in u\circ v\circ w$ .

Hence, the required permutation set is $\left\{1,6,7,4,5,2,3\right\}$ .

(i)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{5,3,7,4,1,2,6\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $u^{-1}$ .

  $u^{-1}=\left\{\left(1,5\right),\left(2,3\right),\left(3,7\right),\left(4,4\right),\left(5,1\right),\left(6,2\right),\left(7,6\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in u^{-1}$ .

Hence, the required permutation set is $\left\{5,3,7,4,1,2,6\right\}$ .

(j)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{5,1,7,3,2,4,6\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $v^{-1}$ .

  $v^{-1}=\left\{\left(1,2\right),\left(2,1\right),\left(3,7\right),\left(4,3\right),\left(5,2\right),\left(6,4\right),\left(7,6\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in v^{-1}$ .

Hence, the required permutation set is $\left\{5,1,7,3,2,4,6\right\}$ .

(k)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{2,7,6,3,5,1,4\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find ${\left(u\circ v\right)}^{-1}$ .

  ${\left(u\circ v\right)}^{-1}=\left\{\left(1,2\right),\left(2,7\right),\left(3,6\right),\left(4,3\right),\left(5,5\right),\left(6,1\right),\left(7,4\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in {\left(u\circ v\right)}^{-1}$ .

Hence, the required permutation set is $\left\{2,7,6,3,5,1,4\right\}$ .

(l)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{2,7,6,3,5,1,4\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $v^{-1}\circ u^{-1}$ .

  $v^{-1}\circ u^{-1}=\left\{\left(1,2\right),\left(2,7\right),\left(3,6\right),\left(4,3\right),\left(5,5\right),\left(6,1\right),\left(7,4\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in v^{-1}\circ u^{-1}$ .

Hence, the required permutation set is $\left\{2,7,6,3,5,1,4\right\}$ .

(m)

To determine

To find: The permutation of the set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Answer to Problem 9E

The required permutation set is $\left\{1,5,6,7,3,4,2\right\}$ .

Explanation of Solution

Given information:

The given set is $C=\left\{1,2,3,4,5,6,7\right\}$ .

Consider the given set $C=\left\{1,2,3,4,5,6,7\right\}$ .

Now, find $u^{-1}\circ v^{-1}$ .

  $u^{-1}\circ v^{-1}=\left\{\left(1,2\right),\left(2,5\right),\left(3,6\right),\left(4,7\right),\left(5,3\right),\left(6,4\right),\left(7,2\right)\right\}$

Here the element $x$ in $C$ is in the place $y$ , if $\left(y,x\right)\in u^{-1}\circ v^{-1}$ .

Hence, the required permutation set is $\left\{1,5,6,7,3,4,2\right\}$ .