# Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that

Question:

Let $A=\{1,2\}, B=\{1,2,3,4\}, C=\{5,6\}$ and $D=\{5,6,7,8\}$. Verify that

(i) $A \times(B \cap C)=(A \times B) \cap(A \times C)$

(ii) $A \times C$ is a subset of $B \times D$

Solution:

(i) To verify: $A \times(B \cap C)=(A \times B) \cap(A \times C)$

We have $B \cap C=\{1,2,3,4\} \cap\{5,6\}=\Phi$

$\therefore$ L.H.S. $=A \times(B \cap C)=A \times \Phi=\Phi$

$A \times B=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)\}$

$A \times C=\{(1,5),(1,6),(2,5),(2,6)\}$

$\therefore$ R.H.S. $=(A \times B) \cap(A \times C)=\Phi$

$\therefore$ L.H.S. $=$ R.H.S

Hence, $A \times(B \cap C)=(A \times B) \cap(A \times C)$

(ii) To verify: $A \times C$ is a subset of $B \times D$

$A \times C=\{(1,5),(1,6),(2,5),(2,6)\}$

$B \times D=\{(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)\}$

We can observe that all the elements of set A × C are the elements of set B × D.

Therefore, A × C is a subset of B × D.