Mathematical Structures

  H6Q1: Let F and G be non-empty families of sets. Consider the following statements: Statement 1: T F and T G are disjoint. Statement 2: There exist A ∈ F and B ∈ G such that A and B are disjoint. (a) Does Statement 1 imply Statement 2? If so, give a proof. Otherwise, give a counterexample. (b) Does Statement 2 imply Statement 1? If so, give a proof. Otherwise, give a counterexample. H6Q2: Prove the following theorem: Theorem: Let a, b, and x be real numbers with a < b. Then (x − a)(x − b) ≥ 0 if and only if x ≤ a or x ≥ b. H6Q3: Suppose A, B, and C are sets. Prove that A ∪ B ⊆ A ∪ C iff B A ⊆ C A. H6Q4: Consider the following statements: Statement 3: Let a, b and n be natural numbers. If a is not a multiple of n and b is not a multiple of n, then ab is not a multiple of n. Statement 4: Let a, b and n be natural numbers. If a is a multiple of n or b is a multiple of n, then ab is a multiple of n. One of these is true and one is false. Prove the correct one and give a counterexample for the other one. H6Q5 (Bonus question): Prove that there are no rational numbers x and y such that x 2 + y 2 = 3. (You can use facts about prime factorizations of integers.)