Q. 1 B4.0( 3 Votes )

For any sets A, B

Answer :

Given: A, B and C three sets are given.


Need to prove: A × (B C) = (A × B) (A × C)


Let us consider, (x, y) A × (B C)


x A and y (B C)


x A and (y B and y C)


(x A and y B) and (x A and y C)


(x, y) (A × B) and (x, y) (A × C)


(x, y) (A × B) (A × C)


From this we can conclude that,


A × (B C) (A × B) (A × C) ---- (1)


Let us consider again, (a, b) (A × B) (A × C)


(a, b) (A × B) and (a, b) (A × C)


(a A and b B) and (a A and b C)


a A and (b B and b C)


a A and b (B C)


(a, b) A × (B C)


From this, we can conclude that,


(A × B) (A × C) A × (B C) ---- (2)


Now by the definition of the set we can say that, from (1) and (2),


A × (B C) = (A × B) (A × C) [Proved]


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