Q1: Determine by using logical equivalences whether the following compound proposition ia a tautology: (pA (p q)) -q Q2: Translate these statements into English, where R(x) is x is a rabbit', and H (x) is x hops" and the domain consists of all animals. a. Vx(R(x) H (x)) c. 3x(R(x)- H (x)) Q3: Use rules of inference to show that the hypotheses "If you send me an email message, then I will finish writing the program," "If you do not send me an email message, then I will go to sleep early," and "If I go to sleep early, then I will wake up feeling refreshed" lead to the conclusion "If I do not finish writing the program, then I will wake up feeling refreshed. Q4: List the members of these sets. a. fxlxis a real number such that -lj b. fx x is a positive integer less than 12) c. fx | x is the square of an integer and x100 d. {x) x is an integer such that x' =2} Q5: Find the truth set of each of these predicates where the domain is the set of integers a. P(x):