Question

Let A = {0, 1, 2, 3, 4}, B= {1, -2, 3, 4, 5, 6} and C = {2, 4, 6, 7}.
(i) Show that $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$.
(ii) Verify this relation using Venn diagram.

Answer 1

First, we find $A\cup (B\cap C)$
Consider $B\cap C$= {1, 2, 3, 4, 5 ,6} $\cap$ {2, 4, 6, 7} = {4,6}
$A\cup (B\cap C)$ = {0,1,2,3,4} U = {4, 6} , = { 0, 1, 2, 3, 4, 6}. (1)
Next, consider $A\cup B$ = {0,1, 2, 3, 4} $\cup$ {1, -2, 3, 4, 5, 6}
= { -2, 0, 1, 2, 3, 4, 5, 6},
$A\cup C$, = {0, 1, 2, 3, 4} $\cup$ {2, 4, 6, 7} = {0, 1, 2, 3, 4, 6, 7}.
Thus, $(A\cup B)\cap (A\cup C)$ = {-2, 0, 1, 2, 3, 4, 5, 6} $\cap$ {0, 1, 2, 3, 4, 6, 7}
= {0, 1, 2, 3, 4, 6}. (2)
From (1) and (2), we get $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
(ii) Using Venn diagram, we have
From (2) and (5) it is verified that $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$.