Binary Operation is an operation defined for any set S such that it takes two elements from S as input and produces a single element in S as output. As the name suggests, binary operations require at least two inputs as it is defined from the cartesian product of set to set itself.
In this article, we will explore binary operations its definition, properties, types of binary operations, and many more. We will also discuss the applications of binary operations and solve some examples on it. Let's start our learning on the topic "Binary Operation".
What are Binary Operations?
Binary operations are the operations that are performed on two inputs. Some fundamental binary operations are addition, subtraction, multiplication, and division. The inputs are known as the operands. Binary operations also have several properties like closure property, associative property, commutative property, identity element, and inverse element.
Binary Operation Definition
Binary operation is defined as the operation on set S which maps the cartesian product of S to the element that belongs to S. Binary operation * on S with elements a and b can be represented as:
* : S × S → S such that for all a, b; a*b ∈ S
\times is a binary operation.
Table of Content
- What are Binary Operations?
- Properties of Binary Operations
- Closure Property in Binary Operations
- Associativity of Binary Operations
- Commutativity of Binary Operations
- Identity Element of Binary Operations
- Inverse Element of Binary Operations
- Types of Binary Operations
- Binary Operation Table
- Applications of Binary Operations
- Binary Operation Examples
- Practice Problems on Binary Operations
Properties of Binary Operations
Binary operations are the operations performed on two elements of a set and the result also belongs to the same set. Some of the properties of the binary operations are:
- Closure Property in Binary Operations
- Associativity of Binary Operations
- Commutativity of Binary Operations
- Identity Element of Binary Operations
- Inverse Element of Binary Operations
Closure Property in Binary Operations
The closure property in binary operation * on set X with element x and y is defined as:
x ∈ X, y ∈ X ⇒ x * y ∈ X
If x and y belong to a set X then the result of the binary operation between them will also belong to the set X
Associativity of Binary Operations
Associativity of binary operation * on set X with element x, y and z is defined as:
(x * y) * z = x* (y * z)
Commutativity of Binary Operations
Commutativity of binary operation * on set X with element x and y is defined as:
x * y = y * x
Identity Element of Binary Operations
Identity element of binary operation * on set X with element x and e is defined as:
x * e = e * x = x
Then, e is called the identity element.
Inverse Element of Binary Operations
Inverse element of binary operation * on set X with element x, y and e is defined as:
x * y = y * x = e
Then x is inverse of y and y is inverse of x.
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Types of Binary Operations
Binary operations are operations which require two inputs. Some of the common types of binary operations are as follows:
- Binary Addition
- Binary Subtraction
- Binary Multiplication
- Binary Division
Let's discuss these common types in detail as follows:
Binary Addition
Binary addition is an operation on a set A with elements x and y defined as:
+ : A × A → A such that (x, y) → x + y
Binary Subtraction
Consider a set A with elements x and y. Binary subtraction is a closed binary operation on A such that:
- : A × A → A such that (x, y) → x - y
Binary Multiplication
Binary multiplication is a binary operation defined on a set A, where each element x and y in A is paired with the operation symbol ×, resulting in x × y, which belongs to the set A. Mathematically this can be written as:
× : A × A → A such that (x, y) → x × y
Binary Division
Binary division on a set A with elements x and y is a binary operation denoted as / and defined as:
/ : A × A → A such that (x, y) → x / y
Binary Operation Table
A binary operation table, also known as a Cayley table or operation table, is a systematic way to display the results of applying a binary operation to elements of a set.
In this table, each row represents one of the elements of the set, and each column represents another element. The cell at the intersection of a row and a column contains the result of applying the binary operation to the corresponding pair of elements.
For example, let's consider a set A={0, 1, 2, 3} with addition modulo(⊕) as operation.
| ⊕ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 1 |
| 2 | 3 | 4 | 1 | 2 |
| 3 | 4 | 1 | 2 | 3 |
| 4 | 1 | 2 | 3 | 4 |
Applications of Binary Operations
Some of the common applications of binary operations are:
- Binary operations are fundamental in abstract algebra, where they are used to define algebraic structures such as groups, rings, and fields.
- In combinatorics, binary operations are used to study various counting problems, permutations, and combinations.
- Binary operations are extensively used in computer science for bitwise operations, such as AND, OR, XOR, and complement operations, which are fundamental in digital logic and computer arithmetic.
- In electrical engineering, binary operations are essential for digital signal processing, coding theory, and error detection/correction techniques.
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Binary Operation Examples
Example: Consider a binary operation * on set X = {1, 2, 3, 4, 5} defined by x*y = min (x, y). With the help of below table find:
(i) Compute (4 * 5) * 1
(ii) Is * commutative?
(iii) Compute (2 * 5) * (1 * 3)
Table:
* | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 2 | 2 | 2 | 2 |
3 | 1 | 2 | 3 | 3 | 3 |
4 | 1 | 2 | 3 | 4 | 4 |
5 | 1 | 2 | 3 | 4 | 5 |
Solution:
(i) (4 * 5) * 1
From table
(4 * 5) = 4
(4 * 5) * 1 = 4 * 1
(4 * 5) * 1 = 1
(ii) Is * commutative
For commutative we have to prove x*y = y*x
let x = 5 and y = 2
x*y = 5 * 2 = 2
y*x = 2 * 5 = 2
Therefore, * is commutative.
(iii) (2 * 5) * (1 * 3)
From table
(2 * 5) = 2
(1 * 3) = 1
(2 * 5) * (1 * 3) = 2 * 1
(2 * 5) * (1 * 3) = 1
Practice Problems on Binary Operations
Problem : Consider a binary operation * on set X = {a, b, c} defined by below. Find:
(i) Compute (a * b) * c
(ii) Is * commutative?
(iii) Find the identity element of the binary operation.
Table:
* | a | b | c |
|---|---|---|---|
a | a | b | c |
b | b | c | a |
c | c | a | b |
Conclusion
Binary operations are essential math ideas used in many areas like algebra, computer science, engineering, and cryptography. Understanding how these operations work and their properties is essential for solving complex problems and building efficient algorithms. This article gave a basic introduction to this important concept including topics such as properties and types of binary operations.