Chapter 6, Problem 1SE

(a)

To determine

To find: The number of system of distinct representatives for the sequence of set $\left\{1,2,3,4,5\right\},\left\{1,2,3,4,5\right\},\left\{1,2,3,4,5\right\}$.

Answer to Problem 1SE

There are $\underset{\_}{60}$ system of distinct representatives for the sequence of set $\left\{1,2,3,4,5\right\},\left\{1,2,3,4,5\right\},\left\{1,2,3,4,5\right\}$.

Explanation of Solution

Given:

The sequence of set is $\left\{1,2,3,4,5\right\},\left\{1,2,3,4,5\right\},\left\{1,2,3,4,5\right\}$.

Concept used:

If the finite sequence of sets $S_1,S_2,\dots ,S_n$ have the elements $x_1,x_2,\dots ,x_n$ then the sets $S_1,S_2,\dots ,S_n$ are the system of distinct representative such that $x_i\in S_i$.

Here, $i$ is equal to $1,2,\dots ,n$ and the element $x_i$ are all distinct.

Calculation:

The set $S_1,S_2,S_3$ can be written as follows:

$\begin{array}{c}{S}_1=\left\{1,2,3,4,5\right\}\\ S_2=\left\{1,2,3,4,5\right\}\\ S_3=\left\{1,2,3,4,5\right\}\end{array}$

To obtain the number of distinct representative, there are 5 ways to choose an element from the set $S_1$ and then 4 ways left to select the element from the set $S_2$ because one element is selected from set $S_1$.

Now, there are only 3 ways to select the element from set $S_3$ for distinct representative.

Therefore, the number $n$ of distinct representative for the sets $S_1,S_2,S_3$ can be calculated as follows:

$\begin{array}{c}n=5\cdot 4\cdot 3\\ =60\end{array}$

Thus, the number of distinct representative for the sets $S_1,S_2,S_3$ is $\underset{\_}{60}$.

(b)

To determine

To find: The number of system of distinct representatives for the sequence of set $\left\{1,2,3,4\right\},\left\{1,2,3,4\right\},\left\{5,6,7\right\}$.

Answer to Problem 1SE

There are $\underset{\_}{36}$ system of distinct representatives for the sequence of set $\left\{1,2,3,4\right\},\left\{1,2,3,4\right\},\left\{5,6,7\right\}$.

Explanation of Solution

Given:

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

Calculation:

The set $S_1,S_2,S_3$ can be written as follows:

$\begin{array}{c}{S}_1=\left\{1,2,3,4\right\}\\ S_2=\left\{1,2,3,4\right\}\\ S_3=\left\{5,6,7\right\}\end{array}$

To obtain the number of distinct representative, there are 4 ways to choose an element from the set $S_1$ and then 3 ways left to select the element from the set $S_2$ because one element is selected from set $S_1$.

Now, there are only 3 ways to select the element from set $S_3$ as $\left\{5,6,7\right\}$ for distinct representative.

Therefore, the number $n$ of distinct representative for the sets $S_1,S_2,S_3$ can be calculated as follows:

$\begin{array}{c}n=4\cdot 3\cdot 3\\ =36\end{array}$

Thus, the number of distinct representative for the sets $S_1,S_2,S_3$ is $\underset{\_}{36}$.

(c)

To determine

To find: The number of system of distinct representatives for the sequence of set $\left\{1,2,3\right\},\left\{2,3,4\right\},\left\{1,2,4\right\},\left\{1,3,4\right\},\left\{2,3,4\right\}$.

Answer to Problem 1SE

There is $\underset{\_}{0}$ system of distinct representatives for the sequence of set $\left\{1,2,3\right\},\left\{2,3,4\right\},\left\{1,2,4\right\},\left\{1,3,4\right\},\left\{2,3,4\right\}$.

Explanation of Solution

Given:

The sequence of set is $\left\{1,2,3\right\},\left\{2,3,4\right\},\left\{1,2,4\right\},\left\{1,3,4\right\},\left\{2,3,4\right\}$.

Concept used:

The sequence of set $S_1,S_2,\dots ,S_n$ is distinct representative if the sequence $x_1,x_2,\dots x_n$ is exists such that $x_i\in S_i$ for $i=1,2,\dots ,n$.

Here, the elements $x_i$ are all distinct.

Calculation:

The sequence of set can be written as follows:

$\begin{array}{c}{S}_1=\left\{1,2,3\right\}\\ S_2=\left\{2,3,4\right\}\\ S_3=\left\{1,2,4\right\}\\ S_4=\left\{1,3,4\right\}\\ S_5=\left\{2,3,4\right\}\end{array}$

The union of set $S_1,S_2\text{,}{S}_3\text{ and }{S}_4$ can be obtained as follows:

$S_1\cup S_2\cup S_3\cup S_4\cup S_5=\left\{1,2,3,4\right\}$

The total number of elements in the above set is 4 such that the elements can be arranged in $4!$ ways that is equal to 24.

The elements $\left\{1,2,3,4\right\}$ can be arranged in 6 ways as follows:

$\begin{array}{l}{S}_1{}^{\prime }=\left\{1,2,3,4\right\}\\ S_2{}^{\prime }=\left\{1,2,4,3\right\}\\ S_3{}^{\prime }=\left\{1,3,2,4\right\}\\ S_4{}^{\prime }=\left\{1,3,4,2\right\}\\ S_5{}^{\prime }=\left\{2,1,3,4\right\}\\ S_6{}^{\prime }=\left\{4,2,3,1\right\}\end{array}$

Similarly, there are 18 more ways to arrange the elements $\left\{1,2,3,4\right\}$.

The sequence of set is distinct representative if the above mentioned sequence satisfies the condition that $x_i\in S_i$.

The general form of the arranged sequences is as follows:

$S^{\prime }=\left\{x_1,x_2,x_3,x_4\right\}$

The set $S_1{}^{\prime }=\left\{1,2,3,4\right\}$ has the values of $x_1,x_2,x_3\text{ and }{x}_4$ are $1,2,3\text{ and 4}$ respectively.

The first element of the set $S_1{}^{\prime }$ is 1 that belongs to set $S_1$, second element is 2 that belongs to set $S_2$, third element is 3 that belongs to set $S_3$ and fourth element is 4 that does not belong to set $S_4$.

Therefore, the set $S_1{}^{\prime }=\left\{1,2,3,4\right\}$ is not distinct representative.

Similarly, all the sets do not satisfy the condition $x_i\in S_i$.

Thus, there is $\underset{\_}{0}$ system of distinct representatives for the sequence of set $\left\{1,4\right\},\left\{2\right\},\left\{2,3\right\},\left\{1,2,3\right\}$.