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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
Proofs:
1. Left as an exercise at the end of this section.
2. Let x f
1
(Y
1
). Is x f
1
(Y
2
)? Since x f
1
(Y
1
), by definition of preimage of a set, f (x) Y
1
.By
hypothesis, Y
1
Y
2
. Thus, f (x) Y
2
. Again, by definition of preimage, x f
1
(Y
2
).
3. Let y f (X
1
X
2
). Then there exists an element x X
1
X
2
such that f (x)=y.Sincex X
1
X
2
,by
definition of intersection, x X
1
and x X
2
. Then f(x) f(X
1
) and f(x) f(X
2
). So, f(x) f(X
1
) f(X
2
).
See the exercises at the end of the section to investigate why the equality does not hold true.
4. We will prove that f
1
(Y
1
Y
2
)=f
1
(Y
1
) f
1
(Y
2
) using two separate parts .
First part: f
1
(Y
1
Y
2
)
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Publisher Resources

ISBN: 9780123822178