Hello, I need help with my mathematical proofs homework.

Please explain in depth and show work, thank you and take your time, no rush.

3.7.1. Given a real number 3:, let A be the phrase \"% Q logically equivalent to (1P] => (IQ] ? Explain. (b) Is the statement (P A Q) V R equivalent to P A (Q V R) ? Explain. (c) Is the statement (P => Q) => R equivalent to P => (Q => R) ? Explain. (d) Is the statement (-IP) (1) (-wQ) the negation of P (:1. Q ? Explain. 3.7.19. Write the statement \"There is no set A, for which A E A\3.7.21. Write the negation of the foowing statements without using the negation symbol ., Also, for each statement, decide whether it is true or false. Explain your answer briey. (a) (V2? 6 W43; 6 W272 > 312) (b) (32: e Z) [(222 = (a: + 1?) => (2:3 e 2)] \f\fQ1. Let P1, P2 and P3 be statements such that P1 A P2, P2 /\\ P3, P1 A P3 are all FALSE. Decide which of the statements below are TRUE. (a) no two of P1,P2, P3 can be true; (b) at least one of P1, P2, P3 must be true; ((3) all of P1, P2, P3 must be faLse. (d) at least one of P1, P2, P3 must be false