   Chapter 7.4, Problem 37ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove that if A and B are any countably infinite set, then A × B is countably infinite.

To determine

To prove:

If A and B are any countably infinite sets, then A×B is countably infinite.

Explanation

Given information:

A and B are any countably infinite sets.

Concept used:

A function is said to be one-to-one function if the distinct elements in domain must be mapped with distinct elements in co-domain.

A function is onto function if each element in co-domain is mapped with atleast one element in domain.

Proof:

Let us consider set A and B as any countably infinite sets.

Since, A is countably infinite set the elements of the set are listed as shown below.

A:a1,a2,a3,........

Similarly, the elements of the set B are listed as shown below

B:b1,b2,b3,........

Now, let us consider the cartesian product of the two set A×B.

The elements of A×B are represented in a grid form as shown below.

(a1,b1)(a1,b2)(a1,b3)(a2,b1)(a2,b2)(a2,b3)(a3,b1) (a3,b2)(a3,b3)

Now, define a function F from Z+ to A×B by starting to count at (a1,b1) and moving horizontally, diagonally and vertically as shown below

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