Binary Operation

A binary operation can be understood as a function f (x, y) that applies to two elements of the same set S, such that the result will also be an element of the set S. Examples of binary operations are the addition of integers, multiplication of whole numbers, etc. A binary operation is a rule that is applied on two elements of a set and the resultant element also belongs to the same set.

In this article, we will understand the concept of a binary operation, its definition, table, and properties. We will also solve a few examples based on binary operation for a better understanding of the concept.

A binary operation on a set is a mapping of elements of the cartesian product set S × S to S, i.e., *: S × S → S such that a * b ∈ S, for all a, b ∈ S. The two elements of the input and the output belong to the same set S. The binary operation is denoted using different symbols such as addition is denoted by +, multiplication is denoted by ×, etc.

The definition of binary operations states that "If S is a non-empty set, and * is said to be a binary operation on S, then it should satisfy the condition which says, if a ∈ S and b ∈ S, then a * b ∈ S, ∀ a, b ∈ S. In other words, * is a rule for any two elements in the set S where both the input values and the output value should belong to the set S. It is known as binary operations as it is performed on two elements of a set and binary means two.

Let us learn about the properties of binary operation in this section. The binary operation properties are given below:

• Closure Property: A binary operation * on a non-empty set P has closure property, if a ∈ P, b ∈ P ⇒ a * b ∈ P. For example, addition is a binary operation that is closed on natural numbers, integers, and rational numbers.
• Associative Property: The associative property of binary operations holds if, for a non-empty set S, we can write (a * b) *c = a*(b * c), where {a, b, c} ∈ S. Suppose Z be the set of integers and multiplication be the binary operation. Let, a = -3, b = 5, and c = -16. We can write (a × b) × c = 240 = a × (b × c). Please note that all binary operations are not associative, for example, subtraction denoted by '-'.
• Commutative Property: A binary operation * on a non-empty set S is commutative, if a * b = b * a, for all (a, b) ∈ S. Suppose addition be the binary operation and N be the set of natural numbers. Let, a = 4 and b = 5, a + b = 9 = b + a.
• Distributive Property: Let * and # be two binary operations defined on a non-empty set S. The binary operations are distributive if, a* (b # c) = (a * b) # (a * c), for all {a, b, c} ∈ S. Suppose * is the multiplication operation and # is the subtraction operation defined on Z (set of integers). Let, a = 3, b = 4, and c = 7. Then, a*(b # c) = a × (b − c) = 3 × (4 − 7) = -9. And, (a * b) # (a * c) = (a × b) − (a × c) = (3 × 4) − (3 × 7) = 12 − 21 = -9. Therefore, a* (b # c) = (a * b) # (a * c), for all {a, b, c} ∈ Z.
• Identity Element: A non-empty set P with a binary operation * is said to have an identity e ∈ P, if e*a = a*e= a, ∀ a ∈ P. Here, e is the identity element.
• Inverse Property: A non-empty set P with a binary operation * is said to have an inverse element, if a * b = b * a = e, ∀ {a, b, e}∈P. Here, a is the inverse of b, b is the inverse of a and e is the identity element.

A binary operation table is a visual representation of a set where all the elements are shown along with the performed binary operation. An example of a binary operation table is shown below where ^ is the binary operation performed on a set S = {1, 2, 3, 4, 5}. Here, ^: S×S→S. Let a be the row elements and b be the column elements, and the operation is defined as a ^ b.

From the given binary operation table, we can clearly see that 1 ^ 1 = 1, 1 ^ 2 = 1, 2 ^ 2 = 2, 3 ^ 4 = 1, and so on. It satisfies the closure property of binary operations as all the output values belong to the given set. Let us see if it satisfies the other properties of binary operations as well or not.

Associative property: Let a=1, b=2, and c=3. Now, a ^ (b ^ c) = 1 ^ (2 ^ 3) = 1 ^ 1 = 1. Let us check the output value of (a ^ b) ^ c.

(1 ^ 2) ^ 3

= 1 ^ 3 (as 1 ^ 2 = 1)

= 1

Therefore, 1 ^ (2 ^ 3) = (1 ^ 2) ^ 3. It satisfies the associative property. You can check it by taking any three values from the given set.

Commutative property: To satisfy the commutative law, the given binary operation table should satisfy the condition that says a ^ b = b ^ a, for all a, b∈S. Let us take a = 3 and b = 4. Now, 3 ^ 4 = 1, and 4 ^ 3 = 1. Therefore, commutative property holds true. The other examples are 1 ^ 2 = 2 ^ 1 = 1, 4 ^ 5 = 5 ^ 4 = 1, and so on.

Identity element: To find the identity element of the given operation, we have to find an element e which satisfies the equation a ^ e = a, for all a∈S. Now, in the given table, if we look carefully, we find that 1 ^ 1 = 1, 2 ^ 1 = 1, 3 ^ 1= 1, 4 ^ 1= 1, and 5 ^ 1 = 1. Therefore, 1 is the identity element.

Inverse property: To find the inverse elements, we have to pair two elements such that a ^ b = b ^ a = e. We already know that e is 1, so the condition is a ^ b = b ^ a = 1. From the table, we find that 1 ^ 2 = 2 ^ 1 = 1. And similarly, 1 ^ 3 = 3 ^ 1 = 1. So, 1 is the inverse of every element in the set.

Important Notes on Binary Operation

• Not all binary operations hold associative and commutative properties.
• A binary operation can be understood as a function f (x, y) that applies to two elements of the same set S, such that the result will also be an element of the set S.

Related Articles on Binary Operation

Check these interesting articles related to the concept of binary operation in math.

• Operations on Sets
• Commutative Property
• Sets
• Properties of Natural Numbers
• Properties of Whole Numbers

What is Binary Operation in Maths?

Binary operations mean when any operation (including the four basic operations - addition, subtraction, multiplication, and division) is performed on any two elements of a set, it results in an output value that also belongs to the same set. If * is a binary operation defined on set S, such that a ∈ S, b ∈ S, this implies a*b ∈ S.

What are the Properties of Binary Operation?

The six properties of binary operations are listed below:

• Closure property
• Commutative property
• Associative property
• Distributive property
• Existence of an identity element
• Inverse property

What is a Binary Operation on a Set?

On a set A, a binary operation * is mapped as (*): A×A→A. This implies that if we perform the given binary operation on any two elements of the set, the output value is also present in the same set. For example, multiplication is a binary operation on a set of natural numbers.

What is Associative Binary Operation?

The associative binary operation satisfies the associative property which states that if a, b, c ∈ S, then a * (b * c) = (a * b) * c, where * is the binary operation defined on a set S. It means that the order in which we are taking elements does not affect the result.

How to Make Binary Operation Table?

To make a binary operation table, follow the steps given below:

• Step 1: Write all the elements of the given finite set in the first row and in the first column.
• Step 2: Perform the binary operation on each of the pair of elements and write the answer in the corresponding cell.

This is how we make or draw a binary operation table.

What is Commutative Binary Operation?

The commutative binary operation satisfies the commutative law which states that if a binary operation * is defined on a set A, then a * b = b * a, for all a,b∈A.

What is Identity Element in Binary Operation?

Generally, the identity element of a binary operation * on a set S is denoted by e such that a * e = e * a = a, for all a ∈ S.

How to Find Identity Element of Binary Operation?

To find the identity element of a binary operation * on a set S, we need to find an element e in S such a*e = e*a = a, for all a ∈ S

How to Find Inverse Element of Binary Operation?

To find the inverse element of a binary operation * on a set S, we need to find an element b in S such a*b = b*a = e, for all a, b ∈ S