Answer :

Given: A, B and C are three given sets


To prove: A (B C) = (A B) (A C)


Let x A (B C)


x A and x (B C)


x A and (x B or x C)


(x A and x B) or (x A and x C)


x A B or x A C


x (A B) ( A C)


A (B C) (A B) ( A C)………(i)


Let y (A B) (A C)


y A B or x A C


(y A and y B) or (y A and y C)


y A and (y B or y C)


y A and y (B C)


y A (B C)


(A B) (A C) A (B C)………(ii)


We know:


P Q and Q P P = Q


From (i) and (ii):


A (B C) = (A B) (A C)


Hence Proved


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