# Homework Help: Prove that X U Y is countable infinite.

1. May 28, 2013

### Nexttime35

1. The problem statement, all variables and given/known data

2. Relevant equations

Countable Infinite is defined if X is infinite and X is isomorphic to the Natural Numbers.

3. The attempt at a solution

Now I assume that XUY is isomorphic to the Natural Numbers. So X ∪ Y ≅ N .

Now here's where I get confused. I am unsure how to define a function that is invertible (to prove the bijection of being 1-1 and onto). Does anyone have an idea on where to go with this proof?
Thank you,
G.

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2. May 28, 2013

### Dick

You don't start by assuming XuY is countable. That's what you want to prove. Suppose X and Y are disjoint. If X ≅ N then there is a bijection f:N->X so you can write X={f(1),f(2),f(3),...}. Similarly Y={g(1),g(2),g(3),...}. Can you define a bijection h:N->XuY? Think of an h mapping even numbers to X and all of the odd numbers to Y.

3. May 28, 2013

### Nexttime35

Ahh, ok, that definitely makes sense. Thank you for the guidance.