1. The problem statement, all variables and given/known data We define a relation ~ for N^2 by: (n, m) ~(k, l) <=> n + l = m + k Show that ~ is a equivalence relation 2. Relevant equations 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 3. 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.