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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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