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    cardinality

    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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  2. #2
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    Quote Originally Posted by dori1123 View Post
    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.
    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 \left| A \right| = \left| B \right| < \infty 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 f:A \mapsto B is bijective.
    From subjectivity, \left( {\forall b \in B} \right)\left( {\exists a \in A} \right)\left[ {f(a) = b} \right] \Rightarrow \left| A \right| \ge \left| B \right|.
    From injectivity, \left( {\forall p \in A} \right)\left( {\exists !q \in B} \right)\left[ {f(p) = q} \right] \Rightarrow \left| B \right| \ge \left| A \right|.
    Therefore \left| B \right| = \left| A \right|.
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