Please I need help with my proofs homework. The instructions are at the end. Thank you!

Evaluated Problems 1 (Sundstrom) Let S be the function that associates with each natural number the set of its natural number divisors. For example, 5(6) 2 {1, 2,3, 6} and 8(10) 2 {1, 2, 5, 10}. a) What is the domain of 8? Determine an appropriate codomain of S. b) Determine S (n) for three prime and four composite values of n. c) Does there exist a natural number n such that 3(7).) has only one element? Explain your reasoning. d) Does there exist a natural number n such that S(n) has exactly two elements? Explain your reasoning. 6) Is the following statement true or false? Explain your reasoning. For all m,n E N, if m % n, then S(m) 79 S(n). f) Is the following statement true or false? Explain your reasoning. For any T Q N, there exists an n E N such that S(n) = T. 4 Let f : S > T be a function. Suppose A and B be subsets of S, and C and D be subsets of T. a) Prove that G g D => f_1[C] Q f_1[D]. Provide a counterexample to show that the converse is false. b) Prove thatf[A] f[B] Q f[A B]. Provide a counterexample to show that f [A B] g f [A] f [B] c) Prove that A Q f'1[f[A]]. Provide a counterexample to show that f'1[f[A]] Q A. Notice that this means, in general, f _1[ f [A]] 7E A. (1) Suppose S = T and that S is the universal set. Show that f [Ac] 79 f [A]6 by showing that neither set is a subset of the other. (Both can be shown with a single counterexample, although this is not required.) 3 Multivariable Functions) Let f : N X Z > Z X Z be given by f(3:, y) = (:1: + 3;, my). a What is f(2,2)? What is f(1,2)? f(2,1)? ( ) b) What is f*1[{(z,1) | z E Z}]? Why? c) Suppose (z, 1) is in the range of f. What can you conclude about 2:? ) d What is f_1[{(z,0) | z E Z}] Why? Includes statement to be proven; begins proof appropriately Missing one of the above Selects the correct proof Selects an technique; correctly inappropriate proof articulates associated technique or wrong assumptions assumptions Proof TechniQUe Mathematical Mathematical content is 1-3 point scale for minor Significant Content correct and relevant mathematical errors mathematical errors Appropriatety concludes proof technique, uses a Missing both of the concluding statement, above and writes \"QED" canclusion Quality appearance. equations wail-aligned. Stylistic aws symbols appropriate Evaluated Problems 1 LetI={27R|0<:r foreachiei di="{(m,y)ERXR|0