Question: "Let A be an uncountable set, B a..."

College
Algebra
Abstract algebra

Ava-May Nelson

Answered question

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.

Answer & Explanation

Nicole Conner

Skilled2021-05-20Added 97 answers

Let $B = b 0, b 1, b 2, . . .$
If C were finite, we would have that $A = B \cup С$ is countable, which is a contradiction. Thus C is infinite, thus it has a countable suset $D \subset C$ .
Let: $D = d 0, d 1, d 2, . . .$
Then we can define a function f: $A \to C$ $f (x) = {d 2, x = b; d 2 + 1, x = d; x, x \in A (B \cup D)$

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