Last modified on August 3rd, 2022

Binary Operation

All the four basic operations we perform, i.e., addition, subtraction, multiplication, and division, are performed on two operands. For example: 2 + 2 = 4, 6 – 3 = 3, 4 × 3 = 12, and 5 ÷ 5 = 1 are performed on two operands.

Even if we consider three or more operands, such as 2 + 2 + 5, we first operate on two operands, 2 + 2 = 4, and then add the result 4 with the third operand to get the result. Such an operation involving two inputs or operands is called a binary operation. The term ‘binary’ means two.

Binary Operation

How to Solve Binary Operations

Let us consider a non-empty set ‘A’ having elements x and y on which the binary operation * is performed. Then, it should satisfy the conditions x ∈ A and y ∈ A, and if x*y = z, then z ∈ A. In other words, the operands and the result must belong to the same set.

There are six main properties followed for solving any binary operation. Let us consider set A discussed above and its elements x, y, and z.

1. Closure Property

A binary operation * on a non-empty set A, where A = {x, y} has closure property, if x ∈ A, y ∈ A ⇒ x * y ∈ A. Thus, addition is a binary operation that is closed for integers (Z), natural numbers (N), and whole numbers (W). It is also closed for real numbers (R) and complex numbers (C).

For example, for set A, if x = 2 ∈ A, y = 3 ∈ A, a * b = (2 * 3) = 6 ∈ A

2. Associative Property

The associative property of binary operations holds if, for a non-empty set A, we can write (x * y) *z = x*(y * z), where A = {x, y, z}. Associate property is also true for addition binary operation. Such that for set A, (x + y) + z = x + (y + z)

For example, for set A, if x = 2 ∈ A, y = 3 ∈ A, and z = 5 ∈ A, then (2 * 3) *5 = 30 = 2*(3 * 5)

Binary operations subtraction and division are not associative.

3. Commutative Property

A binary operation * on a non-empty set A is commutative if x * y = y * x, where (x, y) ∈ A. The commutative property is true for addition and multiplication.

For example, for set A, if x = 2 ∈ A, y = 3 ∈ A, then 2 * 3 = 6 = 3 * 2

Binary operations subtraction and division are not commutative.

4. Distributive Property

Let * and Φ be two binary operations defined on a non-empty set A. The binary operations are distributive if x*(y Φ z) = (x * y) Φ (x * z) or (y Φ z)*x = (y * x) Φ (z * x).

For example, for set A, if x = 2 ∈ A, y = 3 ∈ A, and z = 5 ∈ A, and * is multiplication and Φ is addition, then 2*(3 + 5) = 16 = (2 * 3) + (2 * 5) or (3 + 5)*2 = 16 = (3 * 2) Φ (5 * 2)

Similarly, the binary operations * and – are also distributive for 2*(3 – 5) = -4 = (2 * 3) – (2 * 5) or (3 – 5)*2 -4 = (3 * 2) – (5 * 2)

Binary operations subtraction and division are not distributive.

5. Identity Property

A non-empty set A, with * as the binary operation, is said to hold the identity element, i ∈ A if i * x = x * i= x where ∀ a ∈ P. Thus, if the binary operation * is addition then i = 0 and if * is × then i = 1

For example, for set A, when * is +, and x = 2 ∈ A, then 2 + 0 = 2. Similarly, when * is ×, then 2 × 1 = 2

6. Inverse Property

A non-empty set A, with * as the binary operation, holds the inverse property if x * y = y * x = i, ∀ {x, y, i} ∈ A. Here in set A, x is the inverse of y, y is the inverse of x, and i is the identity element.

Binary Operation Table

It is the tabular representation of all the elements of a set. The binary operation table shows the elements listed along with the binary operation performed.

An example of a binary operation table for set A = {1, 2, 3, 4} is shown below, with # being the binary operation performed. Let x represents the row elements and y the column elements such that the operation is a # b.

#1234
11111
21212
31131
41214

Let us now check whether the above table holds for all the properties of the binary operation.

Closure property: From the table we can see, 1 # 1 = 1, 1 # 2 = 1, 2 # 2 = 2, 3 # 4 = 1, and so on. Thus it satisfies the closure property of binary operation as all outputs also belong to set A like its elements 1, 2, 3, and 4.

Associative property: If x = 1, y = 2, and z = 3, then according to the closure property (x # y)# z = x#(y # z) = (1 # 2)#3 = 1 = 1#(2 # 3) as 1 # 2 = 1 and 1 # 3 = 1. Thus, the above binary operation table satisfies the associative property.

Commutative property: For proving this property, the binary operation table should satisfy the condition x # y = y # x, for all x, y ∈ A. Let us take any two elements from the above table. If x = 3 and y = 4, then 3 # 4 = 1 = 4 # 3. Thus, the above binary operation table satisfies the commutative.

Identity property: To prove the identity property, we have to identify the identity element i, which satisfies the equation x # i = x, for all x ∈ A. For example: 1 # 1 = 1, 2 # 1 = 1, 3 # 1= 1, 4 # 1= 1, and 5 # 1 = 1. Therefore, 1 is the identity element.

Inverse property: For proving the inverse property, we need to prove x # y = y # x = i. As we know, i =1 thus, x # y = y # x = 1. From the table above, 1 # 2 = 2 # 1 = 1 and 1 # 4 = 4 # 1 = 1 and thus 1 is the inverse of every element in the set.

Solved Examples

Show that addition is a binary operation on natural numbers.

Solution:

The set of natural numbers N = {1, 2, 3, 4, 5…..}.
So, every number from 1 to infinity is a natural number.
Now, if we pick any two numbers and add them, for example, 4 and 8 and add them, it will be a natural number here 4 + 8 = 12 ∈ N.
Thus, addition is a binary operation on natural numbers.

Show that binary operation on subtraction is not associative for whole numbers.

Solution:

The set of whole numbers W = {0, 1, 2, 3, 4…..}.
So, every number from 0 to infinity is a whole number.
Now, if we pick any three numbers including 0, for example, 0, 4, and 8, and apply the associate law of subtraction, we get,
(0 – 4) – 8 = 0 – (4 – 8)
12 ≠ 4
Thus, a binary operation on subtraction is not associative for whole numbers

A binary operation table of set X = {a, b, c} is given below. Prove if it satisfies the commutative property. Also, find the identity element.

Solution:

Given, X = {a, b, c}
Now, let us pair any two elements and check if they satisfy binary operations’ commutative property.
Here, we have a ^ b = b and b ^ a = b, b ^ c = a, and c ^ b = a.
So, the given table satisfies the commutative property as x ^ y = y ^ x, for all x, y ∈ X.
Now, for finding the identity element, we should find an element I ∈ X, such that a ^ i = a = i ^ a, for all a ∈ X.
We have a ^ b = b = b ^ a from the table. Also, c ^ a = c = a ^ c.
Therefore, a is the identity element of the given binary operation.

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