Binary Operations

Abstract

Consider the set ℝ of real numbers. We are familiar with the operations of arithmetic, such as addition, subtraction and multiplication, which combine a pair of numbers in ℝ to make a third number, denoted of course by x + y, x − y and xy or x × y. That is, x + y, x − y and x × y have a meaning, for every x, y ∈ ℝ. If we restrict our choice of x and y to the elements of the non—zero reals ℝ*, then x ÷ y also has a meaning for every x, y ∈ ℝ*. The following definition is a generalization of these examples.