Odd numbers and 0

Question

I have two questions:

1. In the script I'm reading right now for discrete mathematics, the definition for an odd number is given by the following equation:

n is odd <=> ∃ an integer k such that n = 2k + 1

My first question is, can you also define it as:

n is odd <=> ∃ an integer k such that n = 2k - 1?

2. One proof for 0 being an even number is that 0 = 2k, where k = 0

But why couldn't you argue that 0 is odd since 0 = 2k + 1, where k = -1/2 ?

EDIT: Ok, I just realized that -1/2 is not an integer...my bad :)