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(am, bm), wherea, 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 aredivisible by 36, where a and b digits.5. Find the least integer n0 satisfying the property that the diophantineequation 4x + 7y = n has a non-negative solution for each n ≥ n0. Prove yourclaim(a) using the principle of mathematical induction;(b) using strong induction.6. For any a ∈ R+, prove that there exist positive integers m, n satisfying1/n < a < m.7. (a) Give an example of a function from P to P that is injective but notsurjective.(b) Give an example of a function from P to P that is surjective but notinjective.8. Consider the following function f : Z → Z wheref(x) = (x + 1 if x is evenx − 1 if x is oddCalculate f ◦ f and decide if f is (i) injective, (ii) surjective .