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CSCI 301, Math Exercises #3

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1. Consider the relation | (divides) on the set Z.
(a) Prove or disprove: | is reflexive.
(b) Prove or disprove: | is symmetric.
(c) Prove or disprove: | is transitive.
2. Assume R and S are two equivalence relations on a set A.
(a) Prove or disprove: R ∪ S is reflexive.
(b) Prove or disprove: R ∪ S is symmetric.
(c) Prove or disprove: R ∪ S is transitive.
3. Consider the function θ : {0, 1} × N → Z defined as θ(a, b) = a − 2ab + b
(a) Prove or disprove: θ is injective.
(b) Prove or disprove: θ is surjective.