Binary operation is an important chapter for both boards and entrance examinations. There are four basic operations namely addition, subtraction, multiplication and division. The main feature of these operations is that when any two numbers \(a\) and \(b\) are given, then we associate the another number as \(a + b\) or \(a – b\) or \(a \times b\) or \(\frac{a}{b}\). 
Binary operation includes two inputs referred to as operands. Binary operation such as addition, multiplication, subtraction, and division take place on two operands. The mathematical procedures that can be done with the two operands are referred to as binary operations.

Binary Operation: Define

A binary operation * on a set \(A\) is a function *∶\(A \times A \to A\). We denote \(*\left( {a,b} \right)\) by \(a*b.\)
In other words, a binary operation (or composition) on a (non-empty set) \(A\) is a rule associated with ordered pair of elements \((a,b)\) of \(A\), some unique element \(a*b\) of \(A\).

Binary Operation: Types

There are four types of binary operations namely:
1. Binary Addition
2. Binary Subtraction
3. Binary Multiplication
4. Binary Division.

(i) Binary Addition

Consider the binary addition on the set of natural numbers \(N\) and real numbers \(R\) if we add two natural numbers \(x\) and \(y\) then the result will also be a natural number. Same rule holds for real numbers also.
\( + :R \times R \to R\) is derived by \((x,y) \to x + y\)
\( + :N \times N \to N\) is derived by \((x,y) \to x + y\)

(ii) Binary Subtraction

Consider the binary subtraction on natural numbers \(N\) and real numbers \(R\). If we subtract two real numbers such as \(x\) and \(y\), the result of this operation will also be a real number. The same rule does not hold for natural numbers because if we take two numbers such as \(x\) and \(y\) and perform binary subtraction on it, then the result may or may not be a natural number.
For example: \(3 – 5 =  – 2\) (\( – 2\) is not a natural number) 
Hence, -: \(R \times R \to R\) is derived by \((x,y) \to x – y\)

(iii) Binary Multiplication

Consider the binary multiplication on natural numbers \(N\) and real numbers \(R\). If we multiply two natural numbers \(x\) and \(y\), then the result will also be a natural number. Same rule holds for real numbers also.
\( \times :R \times R \to R\) is derived by \((x,y) \to x \times y\)
\( \times :N \times N \to N\) is derived by \((x,y) \to x \times y\)

(iv) Binary Division

Consider the binary division on natural numbers \(N\)  and real numbers \(R\). If we divide two real numbers such as \(x\) and \(y\), then the result will also be a real number. Same rule is not true for natural numbers because if we take two numbers such as \(x\)  and \(y\) and perform binary division on it, then the result may or may not be a natural number.
For example: \(1 \div 3 = \frac{1}{3}\) (\(\frac{1}{3}\) is not a natural number)
Hence, \(R \times R\left\{ 0 \right\} \to R\) is derived by \((x,y) \to x \div y,y \ne 0\)

Binary Operation: Examples

Example 1: The operation of addition is a binary operation on the set of natural numbers.
Example 2: The operation of subtraction is a binary operation on the set of integers. But, it is not a binary operation on the set of natural numbers since the subtraction of two natural numbers may or may not be a natural number.
Example 3: The operation of multiplication is a binary operation on the set of natural numbers, set of integers and set of complex numbers.
Example 4: The operation of the set union, intersection and difference is a binary operation on its power set. 

Binary Operation: Properties

The main properties of the binary operations are as follows:

1. Closure Property

Consider a non-empty set \(A\) and a binary operation * on \(A\). Then \(A\)  is closed under the operation *, if \(a \times b \in A\), where \(a\) and \(b\) are elements of \(A\).
Example: The operation of addition on the set of integers is a closed operation.

2. Associative Property

Consider a non-empty set \(A\)  and a binary operation * on \(A\). Then the operation * on \(A\)  is associative, if for every \(a,b,c, \in A\), we have \((a \times b)*c = a*(b \times c)\)

3. Commutative Property

Consider a non-empty set \(A\), and a binary operation * on \(A\). Then the operation * on \(A\)  is associative, if for every \(a,b, \in ,A,\), we have 
\(a \times b = b \times a\)

4. Identity 

Consider a non-empty set \(A\), and a binary operation * on \(A\). Then the operation * has an identity element (if it exists) e in \(A\)  such that
\(a * e = e * a = a\forall a \in A\)

5. Inverse 

Consider a non-empty set \(A\), and a binary operation * on \(A\). Then the operation has the inverse element if for each \(a \in A\), there exists an element \(B\) in \(A\),  such that 
\(a * b\) (right inverse) \( = b * a\) (left inverse) \( = e\), where \(b\) is called an inverse of \(a\).

6. Idempotent 

Consider a non-empty set \(A\), and a binary operation * on \(A\). Then the operation * has the idempotent property if for each a \( \in A\), we have 
\(a*a = a\,\forall a \in A\)

7. Distributivity 

Consider a non-empty set \(A\), and a binary operation * on \(A\). Then the operation * distributes over addition (+), if for every \(a,b,c \in A\), we have
\(a + (b + c) = (a + b) + (a + c)\) [Left distributivity]

\((b + c) + a = (b + a) + (c + a)\) [Right distributivity]

8. Cancellation

Consider a non-empty set \(A\), and a binary operation * on \(A\). Then the operation * has the cancellation property, if for every \(a,b,c \in A\), we have
\(a + b = a + c \Rightarrow b = c\) [Left cancellation]

\(b + a = c + a \Rightarrow b = c\) [Right cancellation]

Binary Operation Table

When the number of elements in a set \(S\) is less, then in that case we can show a binary operation * on the set \(S\) with the help of a table called the binary operation table for the operation *. 
For example, consider a set \(A = \{ 1,2,3\} \) defined by the operation  \(v:R \times R \to R\) such that 
\(\left( {a,b} \right) \to \max \left\{ {a,b} \right\}\)

We can express the following operation table as shown below.

\(\left( v \right)\)	\(1\)	\(2\)	\(3\)
\(1\)	\(1\)	\(2\)	\(3\)
\(2\)	\(2\)	\(2\)	\(3\)
\(3\)	\(3\)	\(3\)	\(3\)

Here,
\(v\left( {1,1} \right) = \left\{ {1,1} \right\} = 1,v = \left( {1,2} \right) = v\left( {2,1} \right) = \left\{ {1,2} \right\} = 2,v\left( {1,3} \right) = v\left( {1,3} \right) = \left\{ {1,3} \right\} = 3,\)
\(v = \left( {2,2} \right) = \left\{ {2,2} \right\} = 2,v\left( {2,3} \right) = v\left( {3,2} \right) = \left\{ {3,2} \right\} = \left\{ {2,3} \right\} = 3,\left( {3,3} \right) = \left\{ {3,3} \right\} = 3\)
Here, we have \(3\) rows and \(3\)  columns in the operation table with \({\left( {i,\,j} \right)^{{\rm{th}}}}\) entry of the table being maximum of \({i^{{\rm{th}}}}\) and \({j^{{\rm{th}}}}\) elements of the set \(A\).
We can generalise it for general operation * : \(A \times A \to A\) If the set \(A = \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\) Then the operation table will have \(n\)  rows and \(n\)  columns with \({\left( {i,\,j} \right)^{{\rm{th}}}}\) entry being \({a_i} \times {a_j}\). 

Conversely, given any operation table having \(n\) rows and \(n\)  columns with each entry being an element of \(A = \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\)we can define a binary operation * : \(A \times A \to A\) given by \({a_i} \times {a_j}\) = the entry in the \({i^{{\rm{th}}}}\)  row and \({j^{{\rm{th}}}}\) column of the operation table. 

Solved Examples – Binary Operation

Q.1.  Consider the binary operation * on \({I^ + }\), the set of positive integers defined by
\(a \times b = \frac{{ab}}{2}\) 
Determine the identity element for the binary operation *, if exists.
Ans:  Let us assume that the identity element \(e\) be a positive integer number, then
\(e * a = a,\,\,a \in {I^ + }\)
\(\frac{{eq}}{2} = a\)
By right cancellation property, we get
\(e = 2\) ….(i)
Similarly, \(a * e = a,\,\,a \in {I^ + }\)
\(\frac{{ae}}{2} = a\)
By left cancellation property, we get
\(e = 2\) ….(ii)
From equation (i) and (ii) for \(e = 2\) we have
\(e * a = a * e = a\)
Therefore, \(2\) is the identity elements for *.

Q.2. Consider the binary operation * on \(Q\) the set of rational numbers, defined by \(a*b = {a^2} + {b^2}\,\forall a,\,b \in Q.\) Determine whether * is commutative
Sol: Let us assume some elements \(a,b, \in Q\), then by definition
\(a * b = {a^2} + {b^2}\)
And \(b * a = {b^2} + {a^2} = a * b\)
Hence, * is commutative.

Q.3. Consider the set \(A = \{  – 1,0,1\} \) Determine whether \(A\) is closed under addition.
Ans: The sum of elements is \(( – 1) + ( – 1) =  – 2\) and \(1 + 1 = 2\) does not belong to \(A\). Hence, \(A\)  is not closed under addition.

Q.4. Consider the binary operation * on \(Q\), the set of rational numbers, defined by \(a * b = a + b – ab\,\forall \,a,\,b\, \in \,Q\) Determine whether * is associative.
Ans: Let us assume some elements \(a,b,c \in Q\), then the definition
\(\left( {a * b} \right) * c = \left( {a + b – ab} \right) * c = \left( {a + b – ab} \right) + c – \left( {a + b – ab} \right)c\)
\(= a + b – ab + c – ca – bc + abc\)
\( = a + b + c – ab – ac – bc + abc\)
Similarly, we have
\(a * \left( {b * c} \right) = a + b + c – ab – ac – bc + abc\)
Therefore, \(\left( {a * b} \right) * c = a * \left( {b * c} \right)\)
Hence, * is associative.

Q.5. Consider the set \(A = \{ 1,2,3\} \) and a binary operation * on the set \(A\) defined by \(a * b = 2a + b\).
Represent operation * as a table on \(A\).
Sol: The table of the operation is shown in fig:

*	\(1\)	\(2\)	\(3\)
\(1\)	\(4\)	\(6\)	\(8\)
\(2\)	\(6\)	\(8\)	\(10\)
\(3\)	\(8\)	\(10\)	\(12\)

Summary

In this article, we have come across the binary operation, its definition, four types of binary operations on sets like binary addition, binary subtraction, binary multiplication, and binary division along with the examples for more understanding of the concept. Also, we have studied the properties of binary operation such as the closure property, the commutative property, the associative property, the distributive property, the identity, the inverse, the idempotent, the cancellation etc. and the binary operation table. For better understanding we have given some solved examples.

Frequently Asked Questions (FAQs) – Binary Operation

Frequently asked questions related to binary operations is listed as follows:

Q.1. Is binary operation a function?
Ans: Yes, a binary operation can be considered as a function whose input is two elements of the same set \(S\) and whose output also is an element of \(S\). Two elements \(a\) and \(b\) of \(S\)  can be written as a pair \(\left( {a,b} \right)\) of elements in \(S\).

Q.2. Is square root a binary operation?
Ans: No.A non-binary operation refers to a mathematical process that only requires one number to achieve something. Addition, subtraction, multiplication, and division are examples of binary operations. Similarly, examples of non-binary operations consist of square roots, factorials, as well as absolute values.

Q.3. How to implement binary operations?
Ans: On the set of real numbers \(R,\,a * b = a + b\) is a binary operation since the sum of two real numbers is a real number. On the set of natural numbers \(N,\,a * b = a + b\) is a binary operation since the sum of two natural numbers is a natural number.

Q.4. What are examples of binary operations?
Ans: Typical examples of binary operations are the addition \(\left(  +  \right)\)  and multiplication \(\left(  \times  \right)\) of numbers and matrices as well as the composition of functions on a singleton set. For example, 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.

Q.5. Explain binary operations.
Ans: A binary operation * on a set A is a function *∶\(A \times A \to A\). We denote \(\left( {a,b} \right)\) by \(\left( {a,b} \right)\).In other words, a binary operation (or composition) on a (non-empty set) A is a rule that associated with every ordered pair of elements \(\left( {a,b} \right)\) of A, some unique element \(a * b\)  of A.