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Question
Let A and B be two sets, each with a 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.
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Solution
A and B are two non empty sets.
Let f be a function from A to B.
It is given that there is injective map from A to B.
That means f is one−one function .
It is also given that there is injective map from B to A .
That means every element of set B has its image in set A.
⇒ f is onto function or surjective.
∴ f is bijective.
(If a function is both injective and surjective, then the function is bijective.)
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