1. Solve:

a) For every integer , prove that 3n - 5 is even if and only if n is odd.

b) Prove the following statement (for every integer 1) using induction. 1 (+1) =1 = +1 .

c) Prove that the following argument is valid. Use only the rules of inference (including the instantiation and generalization rules) and the logical equivalences (as both were presented in class). Be sure to number each line in your proof and label which rules you used to derive it (if it a premise state that it is a premise). () () (() ()) () () () () ()

d) Let () be the predicate "x used a Windows machine, () be the predicate "x uses an Apple machine, and () be the predicate "xs data is secure on their computer. Translate the following expressions into English, simplifying them as much as possible. The universe of discourse is all animals. i) (() ((() ()) ())) . ii) (() (() ()))

2.The ceiling of a real number x, denoted by , is the unique integer that satisfies the inequality: 1 < . In words, is the smallest integer that is greater than or equal to . Answer the next two questions about the ceiling of a number.

a) Compute 0.1,1.7 and (6.3 5.7 + 4.8)

b) Prove or disprove that, for any real number if 1/2 then 2 = 2 1.