BINARY OPERATIONS

Definition: A binary operation maps two elements of a set onto a single element of that set.

Definition: A binary operation * is commutative if x*y = y*x for all values of x and y.

EXAMPLES

Addition, subtraction, multiplication and division of any of the usual sets of numbers are examples of binary operations.
e.g. 3+2 maps two numbers onto the single number 5.
(Note: The "usual" sets of numbers are N, the natural numbers {0, 1, 2, 3, ...}, Z, the integers {0, 1, -1, 2, -2, 3, -3, ...},
  Q, the rational numbers i.e. the set of fractions, R the real numbers, i.e. the rationals and the irrationals, and C the set of complex numbers.)

Note that while addition and multiplication of numbers are commutative, subtraction and division are not.

Taking the average of two rational numbers, a, b i.e. is another binary operation, e.g. in this case .

Addition, subtraction and multiplication of 2x2 matrices gives three more examples of binary operations on our course.

Composition of functions.
For example: Suppose f(x) = 2x+1 and g(x) = 3x²-5. Then:
.

Note: is pronounced "f after g of x". Composition of functions is an important example of a binary operation.

COUNTER EXAMPLES
The square root of a number is not a binary operation since it maps a single number onto a single number.

The average of two integers is not a binary operation since the answer is not always an integer. (As above, where the average of 3 and 2 is 2.5.)

EXERCISE

1. * is a binary operation on the integers Z where x*y = 3x+y.
   (i) Evaluate 2*5
   (ii) Is * a commutative operation on Z?
   (iii) If x*y = x, write y in terms of x.
   Answer
   
2. 
   Answer