I am proving symmetry in a relation.
Assume: I have $a\,R\,b$ which is $x+y\equiv z+w\pmod 2$. I want to show $b\,R\,a$ which would be $z +w\equiv x+y\pmod 2$.
("$x\mid y$" is the divides symbol.)
From the assumption I have $2\mid(z + w) - (x + y)$.
I want to show that $2\mid(x + y) - (z + w)$.
From the assumption can I factor out a $-1$ on top to say.... $2\mid-((x + y) - (z + w))$
and then just say a $-1$ doesn't matter in modular arithmetic? Or does it? My roommate says.. which is true that $-5\bmod 3$ isn't the same as $5 \bmod 3$.
Any help?
Thanks!
Recall that $x\mid y$, the divides relation, says that there is an integer $n$ such that $y=nx$.
Then if $x\mid y$ we have $y=nx$ and $-y=(-n)x$, and $-n$ is also an integer, so $x\mid-y$.
More generally, if $k$ is an integer, then $x\mid y$ implies $x\mid ky$, and this is the special case when $k=-1$.