n-bit messages with a single parity bit added to them

Question

Prove that if n-bit messages have a single parity bit added to them, then the possible encoded messages comprise exactly half of the bit strings of length n+1. Show that the distance between any two of these code words is at least 2.

Answer 1

There are $2^n$ bit strings of length $n$, that is exactly $\frac12$ of $2^{n+1}$. Two words cannot differ only in the parity bit (why?) and two words differing in only one of the first $n$ bits have differnt parity (why?).