- #1

- 1,270

- 0

**1) Let A be an infinite set and B is countably infinite set, prove that |A U B| = |A|.**

__(Incomplete) solution:__

Let B={b_1,b_2,b_3,...} since B is countably infinite.

Take A1 in A such that A1={a_1,a_2,a_3,...}. A1 is countably infinite.

Take A2=A \ A1

So A=A1 U A2

Construct map f: A U B->A such that

f(a_i)=a_(2i), if a_i E A1

f(b_i)=a_(2i-1), if b_i E B

f(x)=x, if x E A2

I think this map is at least onto, but it may not be a one-to-one function. The trouble that I can see is that A1 and B may not be disjoint and A2 and B may not be disjoint. If they overlap, then the map f is not a bijection. How can we fix this problem?

Any help is appreciated!