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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
Theorem 2: Let A be a nonempty set and let be a partition of A. Consider the following relation on A:
aRb if and only if there exists a set S in the partition such that a S and b S.
Then
1. The relation R is an equivalence relation.
2. The equivalence classes determined by R coincide with all the sets of the partition .
Proof:SeeExercis e 23 fortheproofofpart1.ThispartisonlyacheckofthefactthatR is reflexive,
symmetric, and transitive. The three properties of R depend on the three properties of the partition .
Part 2. Consider [a]. By definition
½a= fx A jaRxg= fx A jthere exists a set S
a
in such that x and a are in S
a
g
= fx A jx
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Publisher Resources

ISBN: 9780123822178