# Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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Homework 13 - MATH 310 0101 - Summer I 2010 - Due by Thursday, June 25 For each of the following relations, determine whether it is reﬂexive, symmetric, and/or transitive. In each case either prove it satisﬁes the property or display a counterexample. Use this to determine whether or not each relation is an equivalence relation. 1. Let F ( R ) be the set of real-valued functions on R and let S R . Deﬁne a relation on F ( R ) by f g if and only if f ( x ) = g ( x ) for every x S . 2. Deﬁne another relation on
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Unformatted text preview: F ( R ) by f ∼ g if and only if f ( x ) = g ( x ) for some x ∈ R . 3. The relation < on R . 4. Given a set S , | S | ≥ 2, deﬁne a relation on P ( S ) by A ∼ B if and only if A ∩ B 6 = ∅ . 5. Deﬁne a relation on N by m | n , read “ m divides n ”, if and only if n = k · m for some k ∈ N . 6. Given f : A → B , deﬁne a relation on A by x ∼ y if and only if x and y have the same image under f ....
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## This document was uploaded on 03/25/2011.

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