   Chapter 7.4, Problem 5ES ### 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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# Let 25Z be the set of all integers that are multiples of 25. Prove that 25Z has the same cardinality as 2Z, the set of all even integers.

To determine

To prove:

25Z has the same cardinality as 2Z ,

where 25Z={...,75,50,25,0,25,50,75,....} and 2Z is the set of all even integers.

Explanation

Given information:

25z and 2z

Concept used:

A function is said to be one-to-one function if 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 at least one element in domain.

Proof:

Consider

2Z={...6,4,2,0,2,4,6,....}

And

25Z={...,75,50,25,0,25,50,75,....}

Now in order to show that 2Z and 25Z have the same cardinality, define a bijective map from 2Z to 25Z.

So, consider the map f from 2Z and 25Z defined by, f(2x)=25x where xZ.

Show that f is one to one. For this take 2x1,2x22Z and consider

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