Chapter 7.2, Problem 14ES

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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Explain the mistake in the following “proof.”Theorem: The function f : Z → Z defined by the formula f ( n ) = 4 n + 3 , for each integer n, is one-to-one.“Proof: Suppose any integer n is given. Then by definition of f, there is only one possible value for f ( n ) − namely, 4 n + 3. Hence f is one-to-one.”

To determine

To find out the mistake in the given proof.

Explanation

Given information:

Theorem: The function f:ZZ defined by the formula f(n)=4n+3 for all integers n, is one-to-one.

Proof: Suppose any integer n is given. Then by definition of f, there is only one possible value for f(n), namely 4n+3. Hence f is one-to-one.

Calculation:

The mistake in the proof is that how it can be said that “there is only one possible value for f(n), namely 4n+3 ” without even proving it. Rather proof should have gone in the following way -

Let us suppose x,yZ and assume that f(x)=f(y) and now prove x=y

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