For all proofs in this question, you may use the following fact from a worksheet: x R, 0 x x< 1, (1) and its generalization to the ceiling function: x R, 0 xx < 1. (2)

(a)Prove that x Z, (x-1)/2 = x/2

(b) Prove or disprove each of the following. In each case, first write down in symbolic notation the exact statement you are attempting to prove (either the original statement or its negation). i. x R, x 1= x1 ii. x, y R, xy= xy