1. Construct the truth table for compound proposition (q + p) (p = -9). 2. Express these system specifications using the propositions p "The user enters a valid password," 9 "Access is granted," and r "The user has paid the subscription fee" and logical connectives (including negations). Please note some propositions below may be false. (1) The user has paid the subscription fee but does not enter a valid password. (2) Access is granted whenever the user has paid the subscription fee and enters a valid password. (3) Access won't be denied unless the user has not paid the subscription fee. (4) It's necessary to enter a valid password to have the access grated. 3. Determine whether (p V) ^ Gp Vr) (qVr) is a contingency, a tautology, or a contradiction? 4. P(x,y) means "x + 2y = xy", where x and y are integers. Determine the truth value of: (1) 3yP(3, y) (2) 3xVyP(x,y) (3) Vy3xP(x,y) (4) -Vx3y-P(x,y) 5. (1.7 26) Prove that if n is an integer, then n is even if and only if 7n +4 is even. 5. (1.7 26) Prove that if n is an integer, then n is even if and only if 7n + 4 is even. 6. (1.6 24) Identify the error(s) in the argument that if Vx(P(x) v Q(x)) is true then 3x(P(x) AQ(x)) is true. Steps Argument Vx(P(x) v Q(x)) P(C)vec PC VxP(x) QC VXQ(x) Vx(P(x) v VxQ(x)) Explanation Premise Universal instantiation from (1) Simplification from (2) Universal generalization from (3) Simplification from (2 Universal generalization from (5) Conjunction from (4) and (6)