Prove that X U Y is countable infinite.

[Nexttime35]

Homework Statement

Homework Equations

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

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.

 

[Dick]

Homework Statement

Homework Equations

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

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.

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.

 

[Nexttime35]

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