3. Direct Proofs and Proofs by Contradiction [20 points]

(a) [10 points] Prove directly that:

x R\Z and m Z,

bxc + bm xc = m 1.

Hint: Before considering bm xc carefully write x in terms of its floor and fractional component.

(b) [10 points] Prove by contradiction that: n Z, if n 2 is odd then n is odd.