how to prove: If A and B are finite sets with the same number of elements, then f: A --> B is bijective iff f is injective iff f is surjective.

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- Sep 6th 2008, 02:03 PMdori1123cardinality
how to prove: If A and B are finite sets with the same number of elements, then f: A --> B is bijective iff f is injective iff f is surjective.

- Sep 6th 2008, 02:44 PMPlato
That is a bit confused I think. Because a function is bijective iff it is both injective and surjective. So that is by definition.

Now if then any function from A to B is injective if and only if it is also surjective.

I will help you one way. Suppose that is bijective.

From subjectivity, .

From injectivity, .

Therefore .