Let R be a relation on the set of all integers such that aRb if and only if a + b is even. To show that R is transitive, we must prove that O For any integers a, b and c, if a + b is even and b + c is even, then a + c is even For any integers a, b and c, if a + b is even and b + c is even, then a + c is odd O For any integers a, b and c, if a + b is even or b + c is even, then a + c is even O For any integers a, b and c, if a + b is even or b + c is even, then a +cis odd To show that R is not transitive, we must find a counterexample satisfies which of the following statement? O For some integers a, b and c, a + b is even or b + c is even and a +cis even O For some integers a, b and c, a + b is even or b + c is even and a +cis odd O For some integers a, b and c, a + bi even and b + c is even and a +cis odd For some integers a, b and c, a + b is even and b + c is even and a + c is even

3. Let p(n) = Σª a¡n², where a¿ > 0, be a degree-d polynomial in n, and let k be a constant. Prove the following properties. if k ≥ d, then p(n) = 0(nk); if k ≤d, then p(n) = N(nk); if k = d, then p(n) = 0(nk); if k > d, then p(n) = o(n²); if k < d, then p(n) = w(nk); a. b. C. d. e.

Consider F = {BC -> D, B -> E, CE -> D, E -> CA, BF -> G} and R(A,B,C,D,E,F,G) 1.Find a minimal cover for F. 2.Find all many keys for R.

Let A = {1, 2, 3},         B = {{1}, 2, {1, 3}},         C = {1, {1}, 3}.   Compute the following sets:                                                                         2A∩C − B,         (A ∩ B) × (A − C),         (B ∪ C) − A.

The set of letters S consists of 30 As, 6 Bs, 24 Cs, 15 Ds, and 15 Es. The set of letters T consists of 25 As, 20 Bs, 15 Cs, 5 Ds and 35 Es. What is the mutual index of coincidence between sets S and T? Leave your answer as a fraction in lowest terms

The format P(X) represents as a power set of given "X", (of all X's subsets)   Now, assume there are two sets  A= {q, w, e, r} and B = {w, r, f}. (a) What are the elements of P(A) ∩P(B )?(b) What is the cardinality of P(A ∪B )?(c) What is |P(A) ∪P(B )|?