1. (8 marks) Use the logical equivalences to verify [pAC(pV(pA)p. Supply a reason for each step. 2. (7 marks) Prove that (qA(p-p is a tautology using the logical equivalences. Then, use a truth table to verify it 3. (3+3 marks) Write down the following sentences in predicate logic a) Every positive real number has a unique positive real root. b) The equation x2 -x 1-0 has no solutions over the natural numbers 4. (4 marks) Are the formulas 3(x)(Px) Qand (Vx)Px)-(Bx)Q(x) logically equivalent? Justify your answer by providing a proof that they are logically equivalent or an interpretation in which they have different truth values. 5. (3+3 marks) Write down the negations of the following expressions, so that the symbol does not arise to the left of any quantifier. Indicate whether the negated statement is true. a) Vx E N, x2 N and N 2x 6. (2+ 1 marks) Negate the following sentences a)"Truth is not always popular, but it is always right." b) Some young people are not athletic. 7. (6 marks) Show the following argument is valid. There is an email that is sent but it is not saved in the inbox All emails are saved in the inbox or the inbox is full If the inbox is full, then all emails are lost. Therefore, some email is lost. 8. (5 marks) Prove that nnn+2) is divisible by 9 using mathematical induction. 9. (5 marks) Prove that for all integers n, n2-n3 is odd