Part | Proofs 1. Let R be a relation on Z defined via (a,b) ER 0 Find a non-recursive function g for which (Vn) (ne N) (f(n)=g(n)), and then prove it by induction. 3. Prove by induction that 21=1 San- 1(21 - 1) = (2n - 1)2 4. Let f be a function from Z to Z, where f (n) = in-! if nisodd In +3 if n is even Prove that f is a bijection. 5. Prove that 33 + 2n Part II Proof or Counterexample Determine whether each of the following statements is true or false. Then prove that your response is correct. 1. [(AX B)c (C X D)] [ASC ABCD] -2. (AnB)x (CnD)= (AXC)~(BXD) 3. (AXC)~(B XC)=(AnB)xC 4. [ R is transitive] - [ R-1 is transitive] 5. [ R and S are antisymmetric] - [R US is antisymmetric] 6. [ R and are symmetric] - [R n S is symmetric] [(x) 8) (xA)] - [(x)d) (xA)] = [((x)0 - (x)d)(xA)] -L ( (1'x) d) ( xA) (KE) = ((