Math 290 Final Exam questions with detailed accurate answers and graded A+

Let A be a set and let P(A) denote the power set of A. If |P(AxA)| = 16 then |A|=4. - ANSFalse The statement P : for all integers "x", x0 implies x^2 ≥ 0 is trivially true. - ANSTrue Let a,b,c,d,n be integers. If a+b congruent to c (mod n) and a+d congruent to c (mod n), then b is congruent to d (mod n). - ANSTrue Let n be an integer where n is congruent to 1 mod 4. Then n^12 is congruent to 1 mod 16 - ANSTrue Let P,Q,R be statements. Then Pˇ(QˆR)=(¬(Pˆ¬Q))ˆ(PˇR) - ANSFalse Let a, b be integers. If a^2 is congruent to b^2 (mod 10) then a is congruent to be (mod 10). - ANSFalse The infinite union from n=1 to ∞ (-1/n, 1/n] is (-1,1]. - ANSTrue For all real numbers "x", there is a real number y such that y^3=x. - ANSTrue If |{Ø}| = 0 - ANSFalse There are rational numbers a,b such that a^b is irrational. - ANSTrue Consider the set S= {1/n : n is an integer, n≠0}. which of the following statements about S is true? - ANSf. S has a greatest element and a least element Which of the choices below is the negation of the following statement: there exists n in Natural Numbers, for all x in real numbers, (n=x^2)ˇ(n=x^3) - ANSf. for all n in natural numbers, there exists an x in real numbers, (n≠x^2)^(n≠x^3) Let A = {a,b,c}, B={c,d,e} and C={a,d} be sets which are all subsets of the universal set U={a,b,c,d,e}. Here a,b,c,d,e are distinct elements of U. All but one of the following describe the same set. Which one is different - ANSe. A intersect (B-C) Consider the following theorem and the proposed proof. Theorem: Let x be an integer. Then x^2 is congruent to 1 (mod 3) Note - in these questions if the theorem is false then the proof is false - ANSd. The theorem is false, and the proof is incorrect because it did not cover every case