an injection between two finite sets of the same cardinality is bijective
Lemma.
Let A,B be two finite sets of the same cardinality. If f:A→B is an injective function then f is bijective
.
Proof.
In order to prove the lemma, it suffices to show that if f is an injection then the cardinality of f(A) and A are equal. We prove this by induction on n=card(A). The case n=1 is trivial. Assume that the lemma is true for sets of cardinality n and let A be a set of cardinality n+1. Let a∈A so that A1=A-{a} has cardinality n. Thus, f(A1) has cardinality n by the induction hypothesis. Moreover, f(a)∉f(A1) because a∉A1 and f is injective. Therefore:
f(A)=f({a}∪A1)={f(a)}∪f(A1) |
and the set {f(a)}∪f(A1) has cardinality 1+n, as desired. ∎
Title | an injection between two finite sets of the same cardinality is bijective |
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Canonical name | AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective |
Date of creation | 2013-03-22 15:10:20 |
Last modified on | 2013-03-22 15:10:20 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 6 |
Author | alozano (2414) |
Entry type | Theorem![]() |
Classification | msc 03-00 |
Related topic | SchroederBernsteinTheorem |
Related topic | ProofOfSchroederBernsteinTheorem |
Related topic | OneToOneFunctionFromOntoFunction |