# homework 4 - 4.1/1 8-10 20 21 1 For each of the following...

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4.1 /1, 8-10, 20, 21 1. For each of the following binary relations ρ on , decide which of the given ordered pairs belong to ρ . a. x ρ y x + y < 7; (1, 3), (2, 5), (3, 3), (4, 4) b. x ρ y x = y + 2; (0, 2), (4, 2), (6, 3), (5, 3) c. x ρ y 2 x + 3 y = 10; (5, 0), (2, 2), (3, 1), (1, 3) d. x ρ y y is a perfect square; (1, 1), (4, 2), (3, 9), (25, 5) 8. Let S= {0, 1, 2, 4, 6}. Test the following binary relations on S for reflexivity, symmetry, antisymmetry, and transitivity. a. ρ = {(0, 0), (1, 1), (2, 2), (4, 4), (6, 6), (0, 1), (1, 2), (2, 4), (4, 6)} reflexive, antisymmetric b. ρ = {(0, 1), (1, 0), (2, 4), (4, 2), (4, 6), (6, 4)} symmetric c. ρ = {(0, 1), (1, 2), (0, 2), (2, 0), (2, 1), (1, 0), (0, 0), (1, 1), (2, 2)} symmetric, transitive d. ρ = {(0, 0), (1, 1), (2, 2), (4, 4), (6, 6), (4, 6), (6, 4)} reflexive, symmetric, transitive e. ρ = {} symmetric, antisymmetric, transitive 9. Let S be the set of people in the United States. Test the following binary relations on S for reflexivity, symmetry, antisymmetry, and transitivity. a . x ρ y x is at least as tall as y . reflexive, antisymmetric, transitive b. x ρ y x is taller than y . antisymmetric, transitive c. x ρ y x is the same height as y . reflexive, symmetric, transitive d. x ρ y x is a child of y . antisymmetric e. x ρ y x is the husband of y . antisymmetric, transitive. If x is the husband of y then y is assumed to be a woman, so we never have x ρ y and y ρ x except when x = y f. x ρ y x is the spouse of y . symmetric g. x ρ y x has the same parents as y. reflexive, symmetric, transitive h. x ρ y x is the brother of y. none. not symmetric because y could be the sister of x . 10. Test the following binary relations on the given sets S for reflexivity, symmetry, antisymmetry, and transitivity. a. S= , x ρ y |x| | y| . reflexive, transitive. (not antisymmetric: |-1| |1|, |1| |-1|) b. S= , x ρ y x – y is an integral multiple of 3 reflexive, symmetric, transitive c. S= , x ρ y x y is even symmetric. (not transitive: 3 ρ 8 and 8 ρ 5, but 3 not ρ 5 ) d. S= , x ρ y x is odd transitive e. S= set of all squares in the plane, S 1 ρ S 2 length of side of S 1 = length of side of S 2 reflexive, symmetric, transitive. f. S= set of all finite-length strings of characters, x ρ y number of characters in x = number of characters in y reflexive, symmetric, transitive. g.

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homework 4 - 4.1/1 8-10 20 21 1 For each of the following...

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