If A = {x : x ϵ N, x ≤ 7}, B = {x : x is prime, x < 8} and C = {x : x ϵ N, x is odd

Natural numbers start from 1

$A=\{1,2,3,4,5,6,7\}$

$B=\{2,3,5,7\}$

$C=\{1,3,5,7,9\}$

(i) $B^{\cap} C=\{3,5,7\}$

$\mathrm{AU}\left(\mathrm{B}^{\cap} \mathrm{C}\right)=\{1,2,3,4,5,6,7\}$

$\mathrm{A}^{U} \mathrm{~B}=\{1,2,3,4,5,6,7\}$

$\mathrm{A}^{U} \mathrm{C}=\{1,2,3,4,5,6,7,9\}$

$\left(A^{U} B\right)^{\cap}\left(A^{U} C\right)=\{1,23,4,5,6,7\}$

$\Rightarrow_{A}{ }^{U}{ }_{(B} \cap_{C)}=\left(A^{U}{ }^{B}\right)^{\cap}\left(A^{U} C\right)$

Hence proved

(ii) $\mathrm{B}^{U} \mathrm{C}=\{1,2,3,5,7,9\}$

$A^{\cap}\left(B^{U} C\right)=\{1,2,3,5,7\}$

$A^{\cap} B=\{2,3,5,7\}$

$A^{\cap} C=\{1,3,5,7\}$

$\Rightarrow{ }_{A}^{\cap}\left({ }_{(B} \cup_{C)}={ }_{(A} \cap_{B)} \cup_{(A} \cap_{C)}\right.$