# Proving that AUB is countable Infinite given a specific case

• Jun 3rd 2013, 04:27 PM
nexttime35
Proving that AUB is countable Infinite given a specific case
(a) Prove: If A is a ﬁnite set and B is a countable inﬁnite set and A (intersection) B = null set, then AUB is
countable inﬁnite.
(b) Is 'A (intersection) B = null set ' a necessary assumption?

For A, I understand that I need to find a bijection f:B to Natural numbers, then prove that it's 1-1 and onto by proving its invertiblity.

However, where I get confused is how to incorporate that A is a finite set. After proving that B is countable infinite and that it has a bijection with N, then is there a way to prove that if A is finite, then AUB is also countably infinite?

Thank you.
• Jun 3rd 2013, 04:48 PM
Plato
Re: Proving that AUB is countable Infinite given a specific case
Quote:

Originally Posted by nexttime35
(a) Prove: If A is a ﬁnite set and B is a countable inﬁnite set and A (intersection) B = null set, then AUB is countable inﬁnite.
(b) Is 'A (intersection) B = null set ' a necessary assumption?

I am sure that you understand that any proof depends upon the definitions in use.
The definitions of finite do differ.

But in any case, if $\displaystyle A$ is finite the $\displaystyle \exists k\in\mathbb{N}$ such that $\displaystyle A\sim\{0,\cdots,k\}$

If $\displaystyle A\cap B=\emptyset$ them just shift the map by $\displaystyle k$.

If $\displaystyle A\cap B\ne\emptyset$ them just shift the map by $\displaystyle k-|A\cap B|$.