Recall that a relation on a set, X, is a rule that says whether two elements of that set are "related" in some way.

We write x y if x and y are related by the relation , and x / y if they are not. In general, it depends on the

order we write it in, it may be the case that x y but y / x. For example < is a relation, but 1 < 2, and 2 1.

An equivalence relation, is a relation that is:

reflexive: x x for all x X,

symmetric: if x y, then y x for all x, y X,

transitive: if x y, and y z, then x z for all x, y, z X.

(a) Show that < is not an equivalence relation on R.

(b) Show that equality mod n is an equivalence relation on Z. (recall that x y mod n if x x y is divisible by

n)