Question

Given A = $\{0,1,2,3,4,5\},B=\{0,2,4,6,8\}andC=\{0,3,6,9\}$. Show that

(i) $A\cup (B\cup C)=(A\cup B)\cup C$ i.e. the union of sets is associative.

(ii)$A\cap (B\cap C)=(A\cap B)\cap C$ i.e. the intersection of sets is associative.

Answer 1

A = $\{0,1,2,4,5\}$, B=$\{0,2,4,6,8\}$ and C=$\{0,3,6,9\}$ $B\cup C=\{0,2,4,6,8\}\cup \{0,3,6,9\}=\{0,2,3,4,6,8,9\}$ $\therefore A\cup (B\cup C)=\{0,1,2,4,5\}\cup \{0,2,3,4,6,8,9\}\Rightarrow A\cup (B\cup C)=\{0,1,2,4,5,6,8,9\}...........IA\cup B=\{0,1,2,4,5\}\cup \{0,2,4,6,8\}=\{0,1,2,4,5,6,8\}\therefore (A\cup B)\cup C=\{0,1,2,4,5,6,8\}\cup \{0,3,6,9\}\Rightarrow (A\cup B)\cup C=\{0,1,2,3,4,6,8,9\}.....IIFromIandII,wegetA\cup (B\cup C)=(A\cup B)\cup C$

(ii) $B\cap C=\{0,2,4,6,8\}\cap \{0,3,6,9\}=\{0,6\}Now,A\cup (B\cap C)=\left\{0\right\}..........IA\cap B=\{0,1,2,4,5\}\cap \{0,2,4,6,8\}=\{0,2,4\}\therefore (A\cap B)\cap C=\{0,2,4\}\cap \{0,3,6,9\}\Rightarrow (A\cap B)\cap C=\left\{0\right\}......IIFromIandIIwegetA\cap (B\cap C)=(A\cap B)\cap C$