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Lecture Notes for MATH 3034 - Introduction to Proofs at Virginia Tech (VT)

Notes Information

Material Type:Class Note
Professor:Staff
Class:MATH 3034 - Introduction to Proofs
Subject:Mathematics
University:Virginia Polytechnic Institute And State University
Term:--
Keywords:
  • Provided That
  • Specific Elements
  • Immediately
  • Following Equivalence
  • Circumstances
  • Definitions
  • Nonzero Integers
  • Contradiction
  • Contrapositive
  • Representations
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CHAPTER 5: EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES Section 5.1: Equivalence Relations Relations Examples of relations on the set of real numbers include "=", "<", and "?". Examples of relations on P(R), the power set of R, include "=" and "?". Definition 1:Arelation on a set S is subset of S S. Comments: At first glance, there appears to be a disconnect between the examples of relations given above and the definition of a relation. To make the connection, consider the relation "<"onR. Technically, "<" is a subset of RR. For instance, (1, 2) ?<. Our practice, however, is to write 1 < 2, and we will continue that practice, even in the abstract. If S is a set, we will use the symbol "similarequal" to denote either an abstract relation or a specific relation for which there is no standard notation. For a, b ? S we will write a similarequal b,not (a,b) ?similarequal, to indicate that a and b are related. Definition 2:Letsimilarequal be a relation of a set S.Wesaythatsimilarequal is reflexiv...

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