   Chapter 9.3, Problem 40ES ### 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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# For 40 and 41, use the definition of the Euler phi function ϑ from Section 7.1, exercises 51-53. Use the inclusion/exclusion principle to prove the following: If n = p q , where p and q are distinct prime numbers, then φ ( n ) = ( p − 1 ) ( q − 1 ) .

To determine

To prove the following by usinng the inclusion/exclusion principle: if n=pq, where p and q are distinct prime numbers, then φ(n)=(p1)(q1).

Explanation

Given information:

The inclusion/exclusion principle.

Concept used:

N(AB)=N(U)N(AcBc)

Proof:

The inclusion/Exclusion Rule for Two or three sets.

If A,B and C are any finite sets, then

N(AB)=N(A)+N(B)N(AB)

And

If n=p.q, where p and q are district prime numbers.

By definition of Euler phi function Φ, if p is a prime number and n is an integer with n1, then

Φ(pn)=pnpn1=pn(11p)

Let A and B be the set of all integers that are divisible by P,Q respectively

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