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Proposition: More Characterizations of Finite Sets

Let $X,Y$ be finite sets with equal cardinalities $|X|=|Y|<\infty.$ Then:

1. If $f:X\to Y,$ is surjective, then $f$ is injective.
2. If $f:X\to Y,$ is injective, then $f$ is surjective.

Note: This implies $$f\text{ invective }\Leftrightarrow f\text{ surjective }\Leftrightarrow f\text{ bijective }$$

3 If $|Y| < |X|,$ then no injective function $f:X\to Y$ exists.

3 If $|Y| < |X|,$ then no injective function $f:X\to Y$ exists.

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2

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References

Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001