Question

Recall from class that we defined the set of integers by defining the equivalence relation ∼ on N × N by (a, b) ∼ (c, d) =⇒ a + d = c + b, and then took the integers to be equivalence classes for this relation, i.e. Z = [(a, b)]∼ | (a, b) ∈ N × N . We then proceeded to define 0Z = [(0, 0)]∼, 1Z = [(1, 0)]∼, − [(a, b)]∼ = [(b, a)]∼, [(a, b)]∼ + [(c, d)]∼ = [(a + c, b + d)]∼, and [(a, b)]∼ · [(c, d)]∼ = [(ac + bd, ad + bc)]∼.

1. Show that + is an asociative operation on Z.

2. Show that · is a well-defined operation on Z.

Answer 1