   Chapter 7.4, Problem 12ES ### 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 set of real numbers strictly between 0 and 1. That is S = { x ∈ R|0 x <1 } .Let a and b be real numbers with a < b , and suppose that W = { x ∈ R| a < x < b } . Prove that S and W same cardinality.

To determine

To show:

Sand W have the same cardinality,

where S={x:0<x<1} and W={x:a<x<b}.

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:

It is given that W={x:a<x<b}.

Define f:SW

f(x)=(ba)x+a

To show: f is one-to-one.

Let f(x)=f(y)

i.e. (ba)x+a=(ba)y+a

(ba)x=(ba)yx=y

To show: f is onto

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