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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
To prove that f is onto, let α f1g: Then we can use its prime number factorization and place
all factors with even exponents first, if any. Then
α = α
2c
1
1
α
2c
2
2
α
2c
k
k
α
2c
k+1
1
k+1
α
2c
k+2
1
k+2
α
2c
k+r
1
k+r
:
Then
α = α
2c
1
1
α
2c
2
2
α
2c
k
k
α
2c
k+1
1
k+1
α
2c
k+2
1
k+2
α
2c
k+r
1
k+r
= f ðα
c
1
1
α
c
2
2
α
c
k
k
/α
c
k+1
k+1
α
c
k+2
k+2
α
c
k+r
k+r
Þ= f ðm/nÞ,
with m = α
c
1
1
α
c
2
2
α
c
k
k
and n = α
c
k+1
k+1
α
c
k+2
k+2
α
c
k+r
k+r
: If there are no factors with even exponents, then m = 1:
If there are no factors with odd exponents, then n = 1: So, the function f is one-to-one and onto.
SOLUTIONS FOR THE REVIEW EXERCISES
1. We are assuming that the two points, P and Q are distinct. Therefore, either the values ...
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Publisher Resources

ISBN: 9780123822178