a ) 6) E Proof of P = Q -X= Z A = B -D = -E - Y = Z .'. -BAC= -A C) . ' . -(X A Y ) = Z Proof: F V - D Proof: .. F Proof: Proof of Q = P. IMPORTANT! For each step of your proof, be sure to state the hypothesis, use the hypothesis, and Q2. [2 marks] Let A, B, and C non-empty sets of a universal set U. Give a rigorous state clearly whether it is something you are about to prove or prove. something that follows from a proof of the inclusion of the following sets: previous step or definition. Use appropriate symbols and notations. (AnB) x CE (A x C) U (B x C) Q4. [3 dots] Let U = {0, 2,-1, 6,-5, 4, 3,-3, 1, 5,-6,-2}, A = (2,-1, 4, 3,,-5and-2} , B = Be sure to use proper mathematical notations throughout your Steps. 0, -1, 6,-5, 3,-3. 1} Q3. [4 marks] Give an equivalence proof of the following Theorem 1. Theorem 1: Let A (a) Order the elements of U and B two non-empty subsets of a universal set U. Then (A U B) C BIZZY P (b) Express A and B as a binary string. if and only if A S (An B) Q (c) Find the set defined by the binary string 010100011011. (d) Using binary string operations, determine the binary string that represents A n B and A U B. Q5. [4 marks] Consider the following function:szXRXZ>RXR,where ilq,r.k)=(q-3n -q+r+k) ls finjective? Circle: YES NO Justify your answer with a detailed proof, or with a counterexample and a brief explanation. ls fsurjective? circle: YES NO Justify your answer with a detailed proof, or with a counterexample and a brief explanation, A 06. [4 marks] For this question, you will prove the following equality in two ways: For all A. B and C nonempty U, (A l C) U (Bl C) = C U (B \\A). Q6.a [2 marks] Use a truth table to demonstrate this and be sure to explain clearly how your truth table demonstrates this equality. Q6.b [2 points] Now use set properties to prove it. you should use one (and only one) property at each step. and you should write the name of that property at each step. Don't forget the parentheses. (BONUS) [+2 bonus points] Circle the best answer for each question. You don't need to justify this, but you will only earn bonus points if all your answers are correct. i. For all sets A and B, is the following statement always true, or sometimes false? Statement If A C B, then (A \\ B) S (A x B). Circle: always true sometimes false Statement : If A = BO, then A = B. Circle: always true sometimes false Statement If fis injective, then f(A n B) = f(A) n f(B). Circle: always true sometimes false ii. Let h : R -] - , 1], be a function defined by h(x) = -x2 + 1. Is this h injective? Circle: yes no Is h surjective? Circle: yes no Ish" ({-21) = {- 3 , 3) ? Circle : yes no 6