Question:
If A = {a, b, c, d, e}, B = {a, c, e, g}, and C = {b, e, f, g} verify that:
(i) $A \cap(B-C)=(A \cap B)-(A \cap C)$
(ii) $A-(B \cap C)=(A-B) \cup(A-C)$
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