Q. 235.0( 1 Vote )

Let A and B be two sets each with finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.

Answer :

Given A and B are two finite sets. There are injective maps from both A to B and B to A.

Let f be the injective map defined from A to B.

Thus, we have f is one-one.

We also know that there is a one-one mapping from B to A.

This means that each element of B is mapped to a distinct element of A.

But, B is the co-domain of f and A is the domain of f.

So, every element of the co-domain of the function f has a pre-image in the domain of the function f.

Thus, f is also onto.

Therefore, f is a bijection as it is both one-one and onto.

Hence, there exists a bijection defined from A to B.

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