Chapter 7.4, Problem 28ES

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Chapter
Section

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove that a disjoint union of any finite set and any countably infinite set is countabley infinite.

To determine

To prove:

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

Explanation

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 (AB=0). We then need to proof that AB 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:ABZ+,g(x)={       i            if x=aif(x)+n       if xB

g onto:

Let yZ+.

If yn, then

g(ay)=y

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

y − n > 0).

g(x)=f(x)+n=(yn)+n=y

We then note that for every yZ+, there exists an element z=ayAB or z=xAB (respectively) such that y=g(z)

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