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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
b. Let a : Let [a]
m
be the equivalence class of a with respect to
m
,and½a
s
be the equivalence
class of a with respect to S. Are these two classes equal?
By definition ½a
m
= fn jn
m
ag and ½a
s
= fn jnSag.Letn ½a
m
, or equivalently, let
a
m
n. By Part a. this is e quivalent to stating that nSa. In turn, this is equivalent to saying that
n ½a
s
. Since n ½a
m
if and only if n ½a
s
, the two equivalence classes are identical.
25. a. The proof of this statement has two parts.
First part: If aRb, then aSb.
If aRb, a
4
= b
4
.Thusða + bÞða bÞða
2
+ b
2
Þ= 0. This implies either a = b or a = b.Inanycase,
jaj= jbj. So, aSb.
Second part: If aSb, then ...
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Publisher Resources

ISBN: 9780123822178