Algebra I: A Basic Course in Abstract Algebra

Chapter 2

Binary Operations

We shall now extend the concept of addition and multiplication of numbers to binary operations on other sets, like set of matrices, polynomials, functions, etc. Properties of these binary operations will be studied. Finally, to illustrate this, we shall discuss the symmetries of regular plane figures, for example, the symmetries of an equilateral triangle, square rectangle etc…

2.1 Definition and Examples

The idea of binary operation may be illustrated by the usual operation of addition in ℤ. For every ordered pair of integers (m, n), there is associated an unique integer m + n. We may therefore think of addition as a mapping from ℤ × ℤ into ℤ, where the image of (m, n) ∈ ℤ × ℤ is denoted by m + n. Generalizing this ...