Q. 23.8( 5 Votes )

For any sets A and B, prove that

(A × B) (B × A) = (A B) × (B A).

Answer :

Given: A and B two sets are given.


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


Let us consider, (x, y) (A × B) (B × A)


(x, y) (A × B) and (x, y) (B × A)


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


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


x (A × B) and y (B × A)


(x, y) (A × B) (B × A)


From this, we can conclude that,


(A × B) (B × A) (A B) × (B A) ---- (1)


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


a (A B) and b (B A)


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


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


(a, b) (A × B) and (a, b) (B × A)


(a, b) (A × B) (B × A)


From this, we can conclude that,


(A B) × (B A) (A × B) (B × A) ---- (2)


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


(A × B) (B × A) = (A B) × (B A) [Proved]


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RS Aggarwal - Mathematics