How to prove that a set of numbers is a complete set of units modulo p, where p is prime?

Question

How do I show that if p is prime, then the following

${1, -1, 2, -2,..., \frac{p-1}{2}, -\frac{p-1}{2}}$

is a complete set of units modulo p?

Answer 1

If $x\ne y\in\{1,-1,2,-2,\dots,\dfrac{p-1}2,-\dfrac{p-1}2\}$, then $|x-y|\le|x|+|y|\le2\cdot\dfrac{p-1}2=p-1\implies x\not\equiv y\bmod p$.

Alternatively, if $x\equiv y\bmod p$, then $x\equiv a\bmod p$ has two solutions in $\Bbb Z_p$, a contradiction.