# Algebraic Structure and Properties of Structure | Discrete Mathematics

In this tutorial, we will learn about the algebraic structure, binary operations on a set, and the properties of algebraic structure in discrete mathematics. By Prerana Jain Last updated : May 09, 2023

## Algebraic Structure

A non-empty set **G** equipped with one or more binary operations is said to be an **algebraic structure**. Suppose ***** is a binary operation on **G**. Then **(G, *)** is an **algebraic structure**. **(N,*), (1, +), (1, -)** are all the **algebraic structure**. Here, **(R, +, .)** is an **algebraic structure** equipped with two operations.

## Binary Operation on a Set

Suppose **G** is a non-empty set. The **G X G = {(a,b) : a E G, b E G)}**. If **f : G X G → G** then **f** is called a binary operation on a set **G**. The image of the ordered pair **(a,b)** under the function **f** is denoted by **afb**.

A binary operation on asset **G** is sometimes also said to be the binary composition in the set **G**. If ***** is a binary composition in **G** then, **a * b E G, a, b E G**. Therefore **g** is closed with respect to the composition denoted by *****.

### Example

An addition is a binary operation on the set **N** of natural number. The sum of two natural number is also a natural number. Therefore, **N** is a natural number with respect to addition i.e. **a+b**.

Subtraction is not a binary operation on **N**. We have **4 – 7 = 3** not belong to **N** whereas **4 E N**. thus, **N** is not closed with respect to subtraction, but subtraction is a binary operation on the set of an integer.

## Properties of an Algebraic Structure

By a property of an **algebraic structure**, we mean a property possessed by any of its operations. Important properties of an algebraic system are:

### 1. Associative and Commutative Laws

An operation ***** on a set is said to be associative or to satisfy the associative law if, for any elements **a, b , c** in **S** we have **(a * b) * c = a * (b * c )**

An operation ***** on a set **S** is said to be commutative or satisfy the commutative law if, **a * b = b * a** for any element **a, b** in **S**.

### 2. Identity Element and Inverse

Consider an operation ***** on a set **S**. An element **e** in **S** is called an identity elements for ***** if for any elements **a** in **S - a * e = e * a = a**

Generally, an element **e** is called a left identity or a right identity according to as **e *a** or **a * e = a** where **a** is any elements in **S**.

Suppose an operation ***** on a set **S** does have an identity element **e**. The inverse of an element in **S** is an element **b** such that: **a * b = b * a = e**

### 3. Cancellation Laws

An operation ***** on a set **S** is a said to satisfy the left cancellation law if, **a * b = a * c implies b = c** and is said to satisfy the right cancellation law if, **b * a = c * a implies b = c**

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