Question

Let $A=$ The set of all natural numbers less than $8$, $B=$ The set of all prime Number Less Than $8$, $C=$ The set of even prime number. Verify that $(A\cap B)\times C=(A\times C)\cap (B\times C)$.

Answer 1

Step-1: Given data:

Following are the given sets:

$$\begin{gathered} A=\left\{1,2,3,4,5,6,7\right\} \\ B=\left\{2,3,5,7\right\} \\ C=\left\{2\right\} \end{gathered}$$

Step-2 :Verification of the expression:

$$\begin{gathered} A\cap B=\left\{1,2,3,4,5,6,7\right\}\cap \left\{2,3,5,7\right\} \\ \Rightarrow A\cap B=\left\{2,3,5,7\right\} \\ and(A\cap B)\times C=\left\{2,3,5,7\right\}\times \left\{2\right\} \\ \Rightarrow (A\cap B)\times C=\left\{\left(2,2\right)\left(3,2\right)\left(5,2\right)\left(7,2\right)\right\} \end{gathered}$$

Now,

$$\begin{gathered} (A\times C)=\left\{1,2,3,4,5,6,7\right\}\times \left\{2\right\} \\ \Rightarrow (A\times C)=\left\{\left(1,2\right)\left(2,2\right)\left(3,2\right)\left(4,2\right)\left(5,2\right)\left(6,2\right)\left(7,2\right)\right\} \\ and\left(B\times C\right)=\left\{2,3,5,7\right\}\times \left\{2\right\} \\ \Rightarrow \left(B\times C\right)=\left\{\left(2,2\right)\left(3,2\right)\left(5,2\right)\left(7,2\right)\right\} \\ and(A\times C)\cap \left(B\times C\right)=\left\{\left(1,2\right)\left(2,2\right)\left(3,2\right)\left(4,2\right)\left(5,2\right)\left(6,2\right)\left(7,2\right)\right\}\cap \left\{\left(2,2\right)\left(3,2\right)\left(5,2\right)\left(7,2\right)\right\} \\ \Rightarrow (A\times C)\cap \left(B\times C\right)=\left\{\left(2,2\right)\left(3,2\right)\left(5,2\right)\left(7,2\right)\right\} \end{gathered}$$

Hence,it is verified that $\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}\mathbf{\times }\mathit{C}\mathbf{=}\mathbf{(}\mathit{A}\mathbf{\times }\mathit{C}\mathbf{)}\mathbf{\cap }\mathbf{(}\mathit{B}\mathbf{\times }\mathit{C}\mathbf{)}$