Question

If we are given the sets as $A=\left\{ 1,3,5,7,8 \right\}$ , $B=\left\{ 3, 5, 8 \right\}$ and $C=\left\{ 2,5, 7 \right\}$ , then $\left( A-C \right)\times \left( B-C \right)$ is
1) $\left\{ \left( 1,3 \right),\left( 1,8 \right) \right\}$
2) $\left\{ \left( 1,3 \right),\left( 1,8 \right), \left( 3,3 \right), \left( 3,8 \right), \left( 8,3 \right), \left( 8,8 \right) \right\}$
3) $\left\{\left( 3,8 \right), \left( 8,3 \right), \left( 8,8 \right) \right\}$
4) None of these

Answer 1

Hint: In this problem we need to calculate the value of $\left( A-C \right)\times \left( B-C \right)$ where $A=\left\{ 1,3,5,7,8 \right\}$ , $B=\left\{ 3, 5, 8 \right\}$ and $C=\left\{ 2,5,7 \right\}$. We will first calculate the value of $A-C$ by eliminating the common terms of set $A$ and $C$ from set $A$ and write the remaining values in the set $A$. Similarly we can calculate the value of $B-C$ by using the sets $B$ and $C$. After having the values of $A-C$ and $B-C$ we can calculate the value of $\left( A-C \right)\times \left( B-C \right)$ by forming all possible values in the set.

Complete step-by-step solution:
Given sets are $A=\left\{ 1,3,5,7,8 \right\}$ , $B=\left\{ 3, 5, 8 \right\}$ and $C=\left\{ 2,5,7 \right\}$.
Consider the sets $A=\left\{ 1,3,5,7,8 \right\}$ and $C=\left\{ 2,5,7 \right\}$.
In the above both sets we have the common values $5,7$ .
After eliminating the value $5,7$ from set $A$, the set $A$ will be
$A=\left\{ 1,3,8 \right\}$
According to the definition of the operation $A-C$, the value of $A-C$ can be written as
$A-C=\left\{ 1,3,8 \right\}$
Consider the sets $B=\left\{ 3,5,8 \right\}$ and $C=\left\{ 2,5,7 \right\}$.
In the above both sets we have the common value $5$ .
After eliminating the value $5$ from set $B$, the set $B$ will be
$B=\left\{ 3,8 \right\}$
According to the definition of the operation $B-C$, the value of $B-C$ can be written as
$B-C=\left\{ 3,8 \right\}$
Now the value of $\left( A-C \right)\times \left( B-C \right)$ can be written as
$\left( A-C \right)\times \left( B-C \right)=\left\{ 1,3,8 \right\}\times \left\{ 3,8 \right\}$
Writing all possible combination for the above mentioned relation, then we will have
$\left( A-C \right)\times \left( B-C \right)=\left\{ \left( 1,3 \right),\left( 1,8 \right), \left( 3,3 \right), \left( 3,8 \right), \left( 8,3 \right), \left( 8,8 \right) \right\}$
Hence the option 2 is the correct answer.

Note: The operations on the sets are quite different from the operation on the variables or numerical. In this problem we have to calculate the value of $A-C$, so we have observed red for the common value and eliminated it from the first set which is $A$ and written the remaining elements of the set $A$ as the result of $A-C$. If we have to calculate the value of $C-A$ , then we need to eliminate the common value from set $C$ and write the remaining elements of set $C$ as a result of $C-A$.