if A is infinite and B is a finite subset of
A, then A \ B is infinite∗
mathcam†
2013-03-21 15:54:43
Theorem. If A is an infinite set and B is a finite subset of A, then A \ B
is infinite.
Proof. The proof is by contradiction. If A \ B would be finite, there would
exist a k ∈ N and a bijection f : {1, . . . , k} → A \ B. Since B is finite, there
also exists a bijection g : {1, . . . , l} → B. We can then define a mapping
h : {1, . . . , k + l} → A by
f (i)
when i ∈ {1, . . . , k},
h(i) =
g(i − k) when i ∈ {k + 1, . . . , k + l}.
Since f and g are bijections, h is a bijection between a finite subset of N and
A. This is a contradiction since A is infinite. ∗ hIfAIsInfiniteAndBIsAFiniteSubsetOfAThenAsetminusBIsInfinitei
created: h2013-0321i by: hmathcami version: h34199i Privacy setting: h1i hTheoremi h03E10i
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