So far we have studied sets as unordered collections. However things start getting interesting if we define one or more binary operations on sets. Such operations define structures on sets and we compare different sets in light of their respective structures. Groups are the first (and simplest) examples of sets with binary operations.
A binary operation on a set A is a map from A × A to A. If ◊ is a binary operation on A, it is customary to write a ◊ a′ to denote the image of (a, a′) (under ◊).
For example, addition, subtraction and multiplication are all binary operations on (or or ). Subtraction is not a binary ...