Properties of a Binary Operation: Definition, Theorems

Properties of a binary operation: The binary number system is a variation of the decimal (\(10-\)base) number system. When compared to the decimal system, binary numerals are useful because they make computer and related technology design easier. We have come across various operations like addition, subtraction, multiplication and division of numbers, union and intersection of sets, the composition of functions etc. In all these operations, any two elements of the given set are operated to get a unique element of the same set. This article introduces such operations as functions from the Cartesian product of a set to the set itself.

Properties of Binary Operation: Definition

If \(S\) is a non-empty set, and \(*\) is stated to be a binary operation on \(S,\) then it should meet the condition that says, if \(a∈S\) and \(b∈S,\) then \(a∗b∈S.\) We can define \(*\) as a function from \(S×S\) to \(S\) which associates each ordered pair \((a,b)∈S×S\) to a unique element \(∗(a,b)\) in \(S.\)

\(*:S \times S \to S\)

Instead of writing \(∗(a,b)\) for the image of ordered pair \((a,b) \in S \times S,\) we write it as \(a∗b.\)

In other words, \(*\) is a rule that applies to any two elements in the set \(S\) where both the input and output values must be from the same set. Binary operations are named because they are done on two elements of a set, and binary signifies two.
We use symbols like \(*, \oplus \) and \( \otimes \) and to represent arbitrary binary operations.

Learn about Binary Operations in Detail

Properties of Binary Operations

There properties of binary operations are as follows:

Let \(*\) be the binary operation, and \(S\) be a non-empty set.

1. Closure Property: An operation \(*\) on \(S\) is said to be closed, if \(a∈S, b∈S,\) and \(a∗b∈S.\) For example, natural numbers are closed under the binary operation addition.

2. Commutativity:  If \(a∗b=b∗a\) for all \(a,b∈S\) then a binary operation \(*\) on a set \(S\) is said to be commutative. For example, the binary operations multiplication \((×)\) and addition \((+)\) are commutative on \(\mathbb{Z}\) However, subtraction \((−)\) is not a commutative binary operation on \(Z\) as \(4−2≠2−4.\)

3. Associativity: A binary operation \(*\) on a set \(S\) is said to be associative, if \((a∗b)∗c=a∗(b∗c)\) for all \(a,b,c∈S.\) For example, if \(S\) is a non-empty set, then intersection \((∩)\) and union \((∪)\) are commutative and associative on the power set of \(S,\) as 

\(A∪B=B∪A\)

\(A∩B=B∩A\)

\((A∪B)∪C=A∪(B∪C)\)

\((A∩B)∩C=A∩(B∩C)\) for all \(A,B,C∈P(S)\)

4. Distributivity: Let \(*\) and \(⊕\) be two binary operations on \(S.\) Then \(*\) is said to be distributive over \(⊕,\) if for all \(a,b,c∈S.\)

\(a*(b \oplus c) = (a*b) \oplus (a*c)\) is known as left distributivity of \(*\) over \(⊕.\)

Similarly, \((b \oplus c)*a = (b*a) \oplus (c*a)\) is known as right distributivity of \(*\) over \(⊕.\)

The binary operation multiplication \((∙)\) on \(Z\) is distributive over the binary operation addition \((+)\) on \(\mathbb{Z}\) because 

\((b + c) \cdot a = (b \cdot a) + (c \cdot a)\)

And \(a \cdot (b + c) = (a \cdot b) + (a \cdot c),\) for all \(a,b,c \in Z\)

But, addition \((+)\) is not distributive over multiplication (\(∙)\) because \(2+(4×5)≠(2+4)×(2+5).\)

5. Identity Element:  Let \(*\) be a binary operation on \(S.\) Suppose an element \(e∈S\) exists such that \(a∗e=e∗a=a,\) for all \(a∈S.\) Then, \(e\) is called an identity element for the binary operation \(‘*’\) on sets.

For example: Consider the addition \((+)\) binary operation on \(Z.\) For any \(a∈Z,\) we know that \(a+0=a=0+a.\) Thus, for addition on \(Z, 0\) is the identity element.

Because \(1×a=a=a×1\) for all \(a∈Z,\) the identity element for multiplication on \(Z\) is \(1.\) We know that addition \((+)\) and multiplication \((×)\) are binary operations on \(N\) such that \(n×1=n=1×n\) for all \(n∈N.\) But, there do not exist any natural number \(e\) such that \(n+e=n=e+n\) for all \(n∈N.\)

So, \(1\) is the identity element for multiplication on \(N.\) But, \(N\) does not have identity element for addition on \(N.\)

It follows from the above discussion that a set may or may not have an identity element for a  binary operation defined on it.

Now, a natural question arises: If a set has an identity element for a binary operation defined on it, how many identity elements can it have? 

To answer this question, we have the following theorem:

Theorem:

Let \(*\) be a binary operation on set \(S.\) If \(S\) has an identity element for the binary operation \(*,\) then it is unique.

Proof :  Let \(e_1\)and \(e_2\) be two identity elements for the binary operation \(*\) on \(S.\) Then, 

\(e_1\) is identity element and \(e_2∈S⇒e_1*e_2=e_2……..(i)\)

\(e_2\) is identity element and \(e_1∈S⇒e_1*e_2=e_1……..(ii)\)

From \((i)\) and \((ii),\) we have

\(e_1=e_2\)

Hence, the identity element, if it exists, for a binary operation on a set is unique.

6. Inverse of an Element: Let \(e\) be the identity element in \(S\) for the binary operation \(*\) on \(S.\) If there exists an element \(b∈S,\) such that \(a∗b=e=b∗a.\) Then, the element \(a∈S\) is called an invertible element. The inverse of the element \(a\) is \(b.\)

Thus, an element \(b∈S\) is called an inverse of an element \(a∈S,\) if \(a∗b=e=b∗a\)

Example: Let addition \((+)\) be the binary operation on \(Z.\) We know that the identity element for addition on \(Z\) is \(0.\) Also, 

\(a+(−a)=0=(−a)+a\) for any integer \(a∈ℤ.\)

So, the inverse of \(a∈ℤ\) is \(–a.\)

We know that multiplication is also a binary operation on \(ℤ\) and an identity element for multiplication on \(ℤ\) is \(1.\) But, no element other than \(1∈ℤ,\) is invertible.

Theorem 1:

Let \(‘∗’\) be an associative binary operation on a set \(S\) with the identity element \(e\) in \(S.\) Then, the inverse of an invertible element is unique.

Proof: Let \(a\) be an invertible element in \(S.\)

Let \(b\) and \(c\) be two inverses of \(a∈S\) for the binary operation \(*.\) Then,

\(a*b=b*a=e\) and \(a*c=c*a=e\)

Now, \((b*a)*c=e*c=c\)

And \(b*(a*c)=b*e=b\)

Since \(*\) is an associative binary operation on a set \(S,\) we have

\((b*a)*c=b*(a*c)\)

\(⇒b=c\)

Therefore, \(a\) has a unique inverse.

Remark: The inverse of an element is generally denoted by \({a^{ – 1}}.\) The inverse of an element \(a\) (if it exists) with respect to the addition (or multiplication) binary operations is generally called the additive (or the multiplicative) inverse and is denoted by \(−a\) (or \(\frac{1}{a}\)).

Theorem 2: 

Let \(*\) be an associative binary operation on a set \(S,\) and \(a\) be an invertible element of \(S.\) Then,\({\left( {{a^{ – 1}}} \right)^{ – 1}} = a.\)

Proof: Let \(e\) be the identity element in \(S\) for the binary operation \(*\) on \(S.\) Then,

\(a*{a^{ – 1}} = e = {a^{ – 1}}*a\)

\( \Rightarrow {a^{ – 1}}*a = e = a*{a^{ – 1}}\)

\( \Rightarrow a\) is inverse of \({a^{ – 1}}\)

\( \Rightarrow a = {\left( {{a^{ – 1}}} \right)^{ – 1}}\)

Remark: Let \(*\) be a binary operation on a set \(S\) and \(e\) be the identity element for \(*\) on \(S.\) Then, \(e∗e=e=e∗e.\) This implies that e is invertible and \({e^{ – 1}} = e.\) Thus, the identity element (if it exists), with respect to a given binary operation defined on a set, is always invertible and it is the inverse of itself.

Solved Examples − Properties of a Binary Operation

Q.1. On \({Q_o},\) the set of all non-zero rational numbers, a binary operation \( * \) is defined by \(a*b = \frac{{ab}}{5}\) for all \(a,b \in {Q_0}.\) Find the identity element for \( * \) in \({Q_o}.\) Also, prove that every element of \({Q_o}\) is invertible.
Ans: Let \(e\) be the identity element. Then,
\({\rm{a * e = a = e * a}}\) for all \(a \in {Q_0}\)
\( \Rightarrow \frac{{ae}}{5} = a\) and \(\frac{{ea}}{5} = a\) for all \(a \in {Q_0}\)
\( \Rightarrow e = 5\)
Thus, \(5\) is the identity element for the binary operation \( * \) defined on \({Q_o}.\)
Now,
Let \(x\) be the inverse of an element \(a \in {Q_0}.\) Then,
\(a*x = x*a = e = 5\)
\( \Rightarrow a*x = 5\) and \(x*a = 5\)
\( \Rightarrow \frac{{ax}}{5} = 5\) and \(\frac{{xa}}{5} = 5\)
\( \Rightarrow x = \frac{{25}}{a},\) if \(a \ne 0\)
Thus, every element \(a \in {Q_0}\) is invertible and its inverse is \(\frac{{25}}{a}.\)

Q.2. Let \(A\) be a set having more than one element. Let \(‘∗’\) be a binary operation on \(A\) defined by \(a*b = a\) for all \(a,b \in A.\) Is \(‘∗’\) commutative or associative on \(A\)?
Ans: Let \(a,b \in A.\) Then,
\(a*b = a\) and \(b*a = b\)
Thus, \(a*b \ne b*a\)
So, \(*\) is not commutative on \(A.\)
Let us now check the associativity of \( * \) on \(A.\)
Let \(a,b, c \in A.\) Then,
\((a*b)*c = a*c = a\) and \(a*(b*c) = a*b = a\)
\(\therefore (a*b)*c = a*(b*c)\)
Hence, \( * \) is associative on \(A.\)

Q.3. Find the identity element in the set \({Q^ + }\) of all positive rational numbers for the operation defined by \(a*b = \frac{{ab}}{2}\) for all \(a,b \in {Q^ + }.\)
Ans: Let \(e\) be the identity element in \({Q^ + }.\) Then,
\(a*e = e*a = a\) for all \(a \in {Q^ + }\)
\( \Rightarrow a * e = a\) and \(e*a = a\) for all \(a \in {Q^ + }\)
\( \Rightarrow \frac{{ae}}{2} = a\) and \(\frac{{ea}}{2} = a\) for all \(a \in {Q^ + }\)
\( \Rightarrow e = 2\)
Hence, \(2\) is the identity element in \({Q^ + }.\)

Q.4. Let \(‘∗’\) be a binary operation on \(N\) given by \({\rm{a * b = HCF(a, b)}}\) for all \(a,b \in N,\) Check the commutativity and associativity of \(∗\) on \(N.\)
Ans: Commutativity: For any \(a,b \in N,\) we have
\(a*b = {\mathop{\rm HCF}\nolimits} (a,b) = {\mathop{\rm HCF}\nolimits} (b,a) = b*a\)
Hence, \(*\) is commutative on \(N.\)
Associativity: For any \(a,b,c \in N,\) we have
\((a*b)*c = {\mathop{\rm HCF}\nolimits} (a,b)*c = {\mathop{\rm HCF}\nolimits} (a,b,c)\)
And, \(a*(b*c) = a*{\mathop{\rm HCF}\nolimits} (b,c) = {\mathop{\rm HCF}\nolimits} (a,b,c)\)
\(\therefore (a*b)*c = a*(b*c)\) for all \(a,b,c \in N.\)
Hence, \(‘∗’\) is associative on \(N.\)

Q.5. Let \(‘∗’\) be a binary operation on \(N\) given by \(a*b = {a^b}\) for all \(a,b \in N.\) Is \(‘∗’\) commutative or associative on \(N\)?
Ans: We have
\(2*3 = {2^3} = 8\) and \(3*2 = {3^2} = 9\)
\(\therefore 2*3 \ne 3*2\)
So, \(‘∗’\) is not commutative on \(N.\)
Also, \(2*(2*3) = 2*{2^3} = 2*8 = {2^8} = 256\) and \((2*2)*3 = {2^2}*3 = 4*3 = {4^3} = 64\)
Since \(2*(2*3) \ne (2*2)*3\)
So, \(‘∗’\) is not associative on \(N.\)
Hence, \(‘∗’\) is neither commutative nor associative on \(N.\)

Practice Exam Questions

Summary

A binary operation is a rule that applies to any two elements in \(S\) where both the input and output values must be from the same set. There are many properties of binary operations such as closure, commutativity, associativity, and distributivity. For a non-empty set \(S\) and binary operation \(*,\) if an element \(e∈S\) exists such that \(a*e = e*a = a,\) for all \(a∈S.\) Then, \(e\) is called an identity element for the binary operation \(*\) on \(S.\) An element \(a∈S\) is called an invertible element if there exists an element \(b∈S,\) such that \(a*b = e = b*a\) There are also a few theorems listed with proofs that help us understand the properties of identity and inverse better

Practice Questions on Relations and Functions

Frequently Asked Questions (FAQs)

The most frequently asked questions on Properties of a Binary Operation are answered below:

Q.1. What are binary operations in math?
Ans: If \(S\) is a non-empty set, and \(^ * \) is stated to be a binary operation on \(S,\) then it should meet the condition that says, if \(a∈S\) and \(b∈S,\) then \(a*b∈S.\) \(^*\) is a rule that applies to any two elements in the set \(S\) where both the input and output values must be from the same set.

Q.2. How do you know if an operation is binary?
Ans: Since the sum of two real numbers is a real number,\( f(a,b)=a+b\) is a binary operation on the set of real numbers \(\mathbb{R}.\) As the sum of two natural numbers is a natural number, \(f(a,b)=a+b\) is a binary operation on the set of natural numbers \(\mathbb{N}.\)

Q.3. What are the laws of binary operations?
Ans: The laws of binary operations are as follows:
1. Closure Law
2. Commutative Law
3. Associative Law
4. Identity element
5. Distributive Law
6. Inverse of an element

Q.4. What is an example of a binary operation?
Ans: A binary operation is a function \(f(x,y)\) that is applied to two e of the same set \(S.\) to produce a result also an element of the set \(S.\) The addition of integers and the multiplication of whole numbers are examples of binary operations.

Q.5. What is associative property in binary operations?
Ans: A binary operation \(*\) on \(S\) is said to be an associative binary operation, if \((a*b)*c = a*(b*c)\) for all \(a,b∈S.\)

Q.6.Where can I study Properties of a Binary Operation?
Ans: You can find the most comprehensive and clear description of Properties of a Binary Operation in this article. For further learning and practice download the Embibe app or log in to embibe.com today.

We hope this detailed article on Properties of a Binary Operation helps you. If you have any queries, feel to ask in the comment section below and we will get back to you at the earliest.