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Math 3210-1 HW 4 Due Friday, June 15, 2007 Problem 1 is worth two points, and the others are worth one point each. Set Operations 1. Let {Aj : j ∈ J} be an indexed family of sets and let B be a set. Prove the following generalizations of theorem 6 from class. i T hT = j∈J (B ∪ Aj ). (a) B ∪ A j j∈J i S hS (b) B ∩ j∈J Aj = j∈J (B ∩ Aj ). i T hS (c) B r j∈J (B r Aj ). j∈J Aj = i S hT (d) B r j∈J (B r Aj ). j∈J Aj = Relations 2. Prove or give a counterexample: A × B = B × A. 3. Prove or give a counterexample: (a) (A ∪ B) × C = (A × C) ∪ (B × C). (b) (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D). (c) (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D). 4. Determine which of the three properties (reflexive, symmetric and transitive) apply to each relation. (You can write simply ”R,” ”S,” or ”T”.) (a) Let R be the relation on N given by xRy iff x divides y. (b) Let X be a set and let R be the relation ”′ ⊆ ”‘ defined on subsets of X. (c) Let R be the relation on the real numbers given by xRy iff x − y is rational. (d) Let R be the relation on the real numbers given by xRy iff x − y is irrational. (e) Let R be the relation on the real numbers given by xRy iff (x − y)2 < 0. (f) Let R be the relation on the real numbers given by xRy iff |x − y| ≤ 2. 5. Find examples of relations with the following properties. (a) Reflexive, but not symmetric and not transitive. (b) Symmetric, but not reflexive and not transitive. (c) Transitive, but not reflexive and not symmetric. (d) Reflexive and symmetric, but not transitive. (e) Reflexive and transitive, but not symmetric. (f) Symmetric and transitive, but not reflexive. 6. Let S be the Cartesian coordinate plane R × R and define a relation R on S by (a, b)R(c, d) iff a = c. Verify that R is an equivalence relation and describe a typical equivalence class E(a,b) . 7. Let S be the set Z of all integers and let R = {(m, n) ∈ Z × Z : m − n is even}. Verify that R is an equivalence relation and describe the equivalence class E5 . How many distinct equivalence classes are there?