Prove that a disjoint union of any finite set and any countably infinite set is countabley infinite.

To prove:

A disjoint union of any finite set and any countably infinite set is countably infinite.

Given information:

A disjoint union of any finite set and any countably infinite set.

Proof:

Let A={a1,a2,...,an} be a finite set and let B be a countably infinite set such that A and B are disjoint (A∩B=0). We then need to proof that A∪B is countably infinite.

Since B is countable infinite, B has the same cardinality as Z + and thus there exists a one-to-one correspondence f between B and Z + (which implies that f is onto and one-to-one).

Let us define the function g as:

g:A∪B→Z+,g(x)={       i            if x=aif(x)+n       if x∈B

g onto:

Let y∈Z+.

If y≤n, then

g(ay)=y

If y > n, then there exists some elements x∈B such that f(x)=y−n (as f is one-to-one and

y − n > 0).

g(x)=f(x)+n=(y−n)+n=y

We then note that for every y∈Z+, there exists an element z=ay∈A∪B or z=x∈A∪B (respectively) such that y=g(z)