Question

Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one to one correspondence between A

Abstract algebra
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asked 2021-05-19
Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one to one correspondence between A and C.

Answers (1)

2021-05-20

Let \(B={b0,b1,b2,...}\)
If C were finite, we would have that \(\displaystyle{A}={B}∪С\) is countable, which is a contradiction. Thus C is infinite, thus it has a countable suset \(\displaystyle{D}⊂{C}\).
Let: \(D={d0,d1,d2,...}\)
Then we can define a function f: \(A \rightarrow C\) \(\displaystyle{f{{\left({x}\right)}}}={\left\lbrace{d}{2},{x}={b};{d}{2}+{1},{x}={d};{x},{x}∈{A}{\left({B}∪{D}\right)}\right.}\)

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