   Chapter 7.4, Problem 11ES ### 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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# In 10-14 s denotes the sets of real numbers strictly between 0 and 1. That is S = { x ∈ R|0< x < 1 } Let V = { x ∈ R|2< x < 5 } . Prove that S and V have the same cardinakity.

To determine

To show:

The sets S and V have the same cardinality, where S={x/0<x<1} and V={x/2<x<5}.

Explanation

Given information:

Let S be the set of all real numbers that are between 0 and 1.

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 S be the set of all real numbers which lies between 0 and 1.

That is S={x/0<x<1}.

Let V be the set defined as V={x/2<x<5}.

The objective is to show that S and V have same cardinality.

Define the function f:SV by f(x)=3x+2, for all x in S.

To show: f is one-to-one.

Let x1,x2S such that

f(x1)=f(x2)3x1+2=3x2+23x1=3x2             Subtract 2 from both sides x1=x2                Divide both sides by 3

Therefore, f is one-to-one

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