# 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

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.

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Nicole Conner

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: $$A \rightarrow C$$ $$\displaystyle{f{{\left({x}\right)}}}={\left\lbrace{d}{2},{x}={b};{d}{2}+{1},{x}={d};{x},{x}∈{A}{\left({B}∪{D}\right)}\right.}$$