How to prove the following: Assume A infinite,B denumerable show that A~$A\cup B$

Question

Assume A is infinite,B is denumerable. Prove A$\approx (A \cup B)$(rewrite)

Reference:

7.18 Theorem A is an infinite set iff A is equipotent with a proper subset of itself.

1.20 Theorem If A and B are any classes, then

i) A⊆A∪B and B⊆A∪B.

ii) A∩B⊆A. and A∩B⊆B

My thoughts on it are as follows : By 7.18 if A is infinite then it is equipotent with a proper subset of itself, so by 1.20(i),A$\subseteq A\cup B$

Then there exist an injective function f:$\omega\mapsto A\cup B$ Check f is bijective once defined.

If A is denumerable ,we know there is a bijective function f:$\omega\mapsto A\cup B$ and we have a function g:$A\cup B\mapsto\omega$ Once defined show it is onto

I don’t know how to create brackets for two functions thus I state “once it is created “ Then ?

I don’t think I can conclude somehow?

Answer 1

If $B\subseteq A$, there is nothing to prove, since in that case $A\cup B=A$, so assume that $B\nsubseteq A$. $A$ is infinite, so it has a denumerable subset $D$.

• Show that $B\cup D$ is denumerable.

Let $f:\omega\to D$ and $g:\omega\to B\cup D$ be bijections. Define

$$h:A\cup B\to A:x\mapsto\begin{cases} a,&\text{if }a\in A\setminus(B\cup D)\\ f\left(g^{-1}(x)\right),&\text{if }x\in B\cup D\,, \end{cases}$$

and show that $h$ is a bijection.

(Some of the discussion in my answer to this related question may prove helpful.)