If A = {a, b, c, d, e}, B = {a, c, e, g}, and C = {b, e, f, g} verify that:

Solution:

(i) B - C represents all elements in B that are not in C

$B-C=\{a, c\}$

$A^{\cap}(B-C)=\{a, c\}$

$A^{\cap} B=\{a, c, e\}$

$A^{\cap} C=\{b, e\}$

$\left(A^{\cap} B\right)-\left(A^{\cap} C\right)=\{a, c\}$

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

Hence proved

(ii) $B^{\cap} C=\{e, g\}$

$A-\left(B^{\cap} C\right)=\{a, b, c, d\}$

$(A-B)=\{b, d\}$

$(A-C)=\{a, c, d\}$

$(A-B) \cup_{(A-C)}=\{a, b, c, d\}$

$\Rightarrow A-\left(B^{\cap} C\right)=(A-B) \cup(A-C)$

Hence proved