if you proving someting like if a relation is symetric. say R is a relation on Z and $xRy$ if x+y is a multiple of 3. Then you want to prove symetry and say for some $x,y$ in Z $xRy$ if
$x+y=3k$ k some integer
This implies $y+x=3k$
$yRx$ so $R$ is symetric.
So my question is when i said "some x,y" are we know treating x and y as constants proving what we need to prove and then does this automatically prove its true for all $x,y$ in $Z$? Thanks
You want to prove that
$$(\forall (x,y)\in\Bbb Z^2) \;\; xRy \implies yRx$$
You say :
Let $(x,y)$ be an arbitrary element of $\Bbb Z^2$ such that $xRy.$
$$xRy \implies (\exists k\in\Bbb Z)\;\; :\; x+y=3k$$
$$\implies (\exists k\in\Bbb Z) \; :\; y+x=3k$$ $$\implies yRx.$$ thus $R$ is symetric.