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CS 336 PreTest_sets_3_solutions

# CS 336 PreTest_sets_3_solutions - CS 336 Pre-Test 3...

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CS 336 Pre-Test 3 Solutions Relations and Functions 4. Relations 1) * For each of the following sets, state whether or not it is a partition of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. a) {{0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}} This is a partition b) { , {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}} This is a not partition since it contains the empty set and also has no subset containing 0. c) {{1, 2}, {3, 4}, {5, 6}, {7, 8}, {9, 10}} This is a not partition since it has no subset containing 0. d) {{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8}, {8, 9}, {9, 10}} This is a not partition since it has no subset containing 0 and several elements are contained in distinct subsets. 2) * For each of the following relations, state which of these properties hold: reflexivity, symmetry, transitivity, and antisymmetry. a) “equality” defined on strings This is reflexive(since ( ) ( ) a b a b = = ), symmetric(since ( ) ( ) a b b a = = ), transitive(since ( ) ( ) a b b c a c = = = ), and antisymmetric (since ( ) ( ) a b b a a b = = = ). b) “inequality” defined on strings This is not reflexive (since ( 1 1 ) < ≠< : ), symmetric (since ( ) ( ) a b b a ), not transitive(since ( 1 2 2 1 ) < ≠< ∧ < ≠< but ( 1 1 ) < ≠< : ), and not antisymmetric (since ( 1 2 2 1 ) < ≠< ∧ < ≠< but 1 2 < ≠< ). c) “less than” defined on N (the natural numbers) This is not reflexive (since (0 0) < : ), , not symmetric (since (0 1) < but (1 0) < : ), transitive (since ( ) ( ) a b

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