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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
Let x ∈⋃
f F
ðB A
f
Þ: Then, by definition of union of a family of sets, x is an element of at least one of the
sets that form the union. So, x B A
f
for at least one f F, call tf
1
. Then x B A
f
1
. By definition of
intersection, this implies that x B and x A
f
1
. The fact that x A
f
1
implies that x is an element of the
union of A
f
1
with any other set. Thus, x ð
f F
A
f
Þ: Also x B . So, x B ð
f F
A
f
Þ: This proves that
B ð
f F
A
f
Þ⊇⋃
f F
ðB A
f
Þ: The two inclusions prove that the sets are equal.
See the set of exercises below for the second proof.
Exercises
18. Consider the family {A
n
for n } with {A
n
=[n, n]={x |n x n}. Find ...
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Publisher Resources

ISBN: 9780123822178