Q. 44.3( 6 Votes )

(i) If A B, prove that A × C B × C for any set C.

(ii) If A B and C D then prove that A × C B × D.

Answer :

(i) Given: A B


Need to prove: A × C B × C


Let us consider, (x, y) (A × C)


That means, x A and y C


Here given, A B


That means, x will surely be in the set B as A is the subset of B and x A.


So, we can write x B


Therefore, x B and y C (x, y) (B × C)


Hence, we can surely conclude that,


A × C B × C [Proved]


(ii) Given: A B and C D


Need to prove: A × C B × D


Let us consider, (x, y) (A × C)


That means, x A and y C


Here given, A B and C D


So, we can say, x B and y D


(x, y) (B × D)


Therefore, we can say that, A × C B × D [Proved]


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RS Aggarwal - Mathematics