Binary Operation - Properties and Examples

Properties and Examples

Typical examples of binary operations are the addition (+) and multiplication (×) of numbers and matrices as well as composition of functions on a single set. For instance,

• On the set of real numbers R, f(a,b) = a + b is a binary operation since the sum of two real numbers is a real number.
• On the set of natural numbers N, f(a,b) = a + b is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.
• On the set M(2,2) of 2 × 2 matrices with real entries, f(A, B) = A + B is a binary operation since the sum of two such matrices is another 2 × 2 matrix.
• On the set M(2,2) of 2 × 2 matrices with real entries, f(A, B) = AB is a binary operation since the product of two such matrices is another 2 × 2 matrix.
• For a given set C, let S be the set of all functions h: C → C. On S, f(g,h) = g ◌ h = g(h(c)), the composition of the two functions g and h, is a binary operation since the composition of the two functions is another function on the set C (that is, a member of S).

Many binary operations of interest in both algebra and formal logic are commutative, satisfying f(a,b) = f(b,a) for all elements a and b in S, or associative, satisfying f(f(a,b), c) = f(a, f(b,c)) for all a, b and c in S. Many also have identity elements and inverse elements.

The first three examples above are commutative and all of the above examples are associative.

On the set of real numbers R, subtraction, that is, f(a,b) = a - b, is a binary operation which is not commutative since, in general, a - b ≠ b - a.

On the set of natural numbers N, the binary operation exponentiation, f(a,b) = ab, is not commutative since, in general, ab ≠ ba and is also not associative since f(f(a,b),c) ≠ f(a, f(b,c)). For instance, with a = 2, b = 3 and c = 2, f(23,2) = f(8,2) = 64, but f(2,32) = f(2,9) = 512. By changing the set N to the set of integers Z, this binary operation becomes a partial operation since it is now undefined when a = 0 and b is any negative integer. For either set, this operation has a right identity (which is 1) since f(a, 1) = a for all a in the set, which is not an identity (two sided identity) since f(1, b) ≠ b in general.

Division (/), a partial operation on the set of real or rational numbers, is not commutative or associative as well. Tetration(↑↑), as a binary operation on the natural numbers, is not commutative nor associative and has no identity element.