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1. Prove that for any sets A and B, (A ∪ B) \ (A ∩ B) = (A \ B) ∪ (B \ A). 2. State and prove a relationship between gcd(a, b) and gcd(a m, bm), where a, b, and m are positive integers, each greater than 1. 3. Let a and b be positive rational numbers. (a) Prove that if √ a + √ b is rational, then √ a − √ b must also be rational. (b) Is the converse of the result in part (a) true? Explain your answer. 4. (a) State and prove a characterization for an integer to be divisible by 36. (b) Use part (a) to find all five digits numbers of the form 1 9 a 9 b that are divisible by 36, where a and b digits. 5. Find the least integer n0 satisfying the property that the diophantine equation 4x + 7y = n has a non-negative solution for each n ≥ n0. Prove your claim (a) using the principle of mathematical induction; (b) using strong induction. 6. For any a ∈ R +, prove that there exist positive integers m, n satisfying 1/n < a < m. 7. (a) Give an example of a function from P to P that is injective but not surjective. (b) Give an example of a function from P to P that is surjective but not injective. 8. Consider the following function f : Z → Z where f(x) = ( x + 1 if x is even x − 1 if x is odd Calculate f ◦ f and decide if f is (i) injective, (ii) surjective .
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