Binary Operations | Properties of Binary Operation

When two numbers are either added, subtracted, multiplied, or divided. The binary operations associate any two elements of a set. the resultant two are in the same set.

An operation that is performed on a single number yields another number of the same or a different system is called a unary operation.

Example of unary operations are negation of a given number, extraction of square roots or cube roots of the number, squaring a number or raising it to a higher power.

We now consider the binary operation of much greater importance, operation which requires two numbers. We start by giving a formal definition of such an operation.

A binary number denoted as ⊗ read  as star  on a non-empty set G is a function which associates with each ordered pair (a, b) of element of G, a unique element, denoted as a ⊗ b of G.

In other words, a binary operation on a set G is a function from the set G x G to the set G. For convenience we often omit the word binary before operation. Also in place of saying ⊗ is an operation on G, we shall say G is closed with respect to ⊗.

Example

Ordinary addition, multiplication are operations on N. i.e. N is closed with respect to ordinary addition and multiplication because

∀ a, b∈ N, a + b ∈ N∧ a.b ∈ N

(∀ stands for “for all” and ∧ stands for “and”)

Example

With obvious modification of the meanings of the symbols, let E be any even natural number and O be any odd natural number, then

E ⨁ E = E (Sum of two even number is an even number)

E ⨁ O = O

And

O ⨁ O = E

These results can be beautifully shown in the form of a table given above.

This shows that the set {E, O} is closed under (ordinary) addition.

The table may be read (horizontally)

E ⨁ E = E           E + E = E

O ⨁ O = O         O + O = O

Example

It can be easily verified that ordinary multiplication (But not addition) is an operation on the set {1, w, w² } where w³ = 1. The adjoining table may be used for the verification of this fact.

(w is pronounced omega)

Properties of Binary Operation

Let S be a non-empty set ⊗ a binary operations on it. Then ⊗ may possess one or more of the following properties.

Commutativity of Binary Operation

* is said to be commutative if

a * b = b * a  ∀ a, b ∈ S

Associativity of Binary Operation

* is said to be associative if

a * (b * c) = (a * b) * c  ∀ a, b, c ∈ S

Existence of an identity element

An element e ∈ S is called an identity element w.r.t * if

a * e = e * a = a, ∀ a ∈ S

Existence of inverse of each element

For any element a ∈ S, ∃ an element a’ ∈ S such that

a * a’ = a’ * a = e  (the identity element)

The symbol ∃ stands for “there exist”.

Theorem of Binary Operations

1. In a set having a binary operation * a left identity and a right identity are the same.

2. In a set having an associative  binary operation left inverse of an element is equal to its right inverse.

Proof

1. Let e’ be the left identity and e” be the right identity. Then

e’ * e” = e’ (e” is a right identity)

            = e’ (e’ is a left identity)

Hence  e’ = e” = e

Therefore, e is the unique identity of S under *

2. For any a ∈ S, let a’, a” be its left and right inverse respectively then a’ * (a * a”) = a’ * e      (a” is right inverse of a)

                                                        = a’    (e is a left identity)

Also (a’ * a”) * a” = e * a”    (a” is left inverse of a)

But a’ * (a’ * a”) = (a’ * a) * a” (* is associative as supposed)

a’ = a”

Inverse of a is generally written as a^-1.

Frequently Asked Question-FAQs