3.6 Structure

I took some time to show the operations and the properties of the real numbers and its subsets like the integers and rational numbers because these are very common in other parts of mathematics when properly abstracted. This structure allows us to learn and prove things and then apply them to new mathematical collections as we encounter them. We start with three: groups, rings, and fields.

3.6.1 Groups

Consider a collection of objects which we call G. For example, G might be Z, Q, or R as above. We also have some pairwise operation between elements of G we denote by ‘‘○’’. It’s a placeholder for an action that operates on two objects.

This ‘‘○’’ operation could be addition ‘‘+’’ or multiplication ‘‘×’’ for numbers, but might be something entirely different. Use your intuition with numbers, but understand that the general...