22 PS4 - Handout #22 April 19, 2010 CS103 Robert Plummer...

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Handout #22 CS103 April 19, 2010 Robert Plummer Problem Set #4—Due Monday, April 26 in class 1. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric and/or transitive, where (x,y) is in R if and only if: a) x = y+1 or x = y-1 b) x is a multiple of y c) x and y are both negative or both nonnegative d) x = y 2 e) x y 2 2. Suppose that R and S are reflexive relations on a set A. Prove or disprove each of the following statements: (a) R S is reflexive (b) R S is reflexive (c) R – S is irreflexive 3. It was noted in Handout #17 that an equivalence relation partitions a set into disjoint, non-empty subsets (p. 5). Those subsets are called equivalence classes, as defined in class on 4/19. Suppose R is an equivalence relation on set A, and that a and b are elements of A. Prove that the following three statements are equivalent: (i) aRb (ii) [a] = [b] (iii) [a] [b] 4. Let R 1 and R 2 be equivalence relations on A. Show that R1
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22 PS4 - Handout #22 April 19, 2010 CS103 Robert Plummer...

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