Section5.3Properties of Congruence

There are two main sets of propositions that make this possible. The proofs are not hard, and you may skip them on a first reading.

Combine this with what one should already know about equivalence relations, and we are ready to roll. Why?

The relation properties mean that we can divide up all integers into "equivalence classes". That is, \(\mathbb{Z}\) can be broken up into disjoint pieces which we are free to consider as units, and which stay different.

The fact that equivalence classes for equivalence relations partition the set is assumed as background knowledge. As for well-definedness, it means that if I want to do a computation, I can pick any number with the same remainder modulo \(n\), and it will still work fine. (Hopefully I pick an easier number to work with!) Here is an example.

Example5.3.3

For instance, \(2\equiv 5\) mod (\(n\)) is the same thing as saying \(5\equiv 2\) mod (\(n\)), and if \(2\not\equiv 6\) mod (\(n\)), then \(5\not\equiv 6\) mod (\(n\)) either.

Or instead of computing \(2\cdot 2\cdot 2\cdot 2\) modulo \(3\), I might choose \(-1\cdot -1\cdot -1\cdot -1\) instead.

It won't always be that clear-cut, but that is the general idea.