We define a relation ~ for N^2 by:
(n, m) ~(k, l) <=> n + l = m + k
Show that ~ is a equivalence relation
A relation R on a set A is equivalent if R is:
reflexive if x R x for all x that is an element of A
symmetric if x R y implies y R x, for all x,y that is an element of A
transitive if x R y and y R z imply x R z, for all x,y,z that is an element of A
The Attempt at a Solution
I have to prove this by showing that the relation is reflexive, symmetric and transitive. The symmetric part i understand, the fact that we can turn the relation around and it will be the same: n + l = m + k vs m + k = n + l.
But how do I prove the refexive and transitive part? I am really confused by the ~ and do not quite understand why it appears both as the name of the relation and as an element of the relation itself.