Notes on Introduction,Binary Operation and its Properties | Grade 12 > Mathematics > Elementary Group Theory | KULLABS.COM

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#### Introduction

A Set is one of the most important concepts in mathematics. It alone conveys very limited information. It becomes more meaningful if we can relate or do something with or operate on its members or elements. The simple way is to relate or combine an element of a set with itself and get the same or a different element of the same set.For instances 1 multiplied by 1 yields 1 but 1 added to 1 yields 2.a different number. We may also relate or combine an element of a set with a different element of the set and arrive at one of the numbers or completely different number. For example 1 multiplied by 2 yields 2 but 1 added to 2 yields 3, a completely different number. Note that, in all cases, the resulting number belongs to the same original set or set under reference i.e. integer Z.

If any two elements or repeated consideration of the same element of the set gives rise to an element of the same set under a given operation, such an operation is known as a binary operation. A set together with a binary operation gives rise to what is known as an algebraic structure or system. An algebraic structure can be realised more correctly once some other characteristic of the operation (+, -, x . .) are specified. Having done so we arrive at the starting point of what is known as group theory. Group theory originated from the study of polynomial equations around the 1830s. three pioneers of Group theory are Galis, A. Cayley and S. Lie.

#### Binary Operation on Sets of Integers

Counting numbers 1, 2, 3, 4, . . . . the number zero and the negatives of the counting numbers -1, -2,- 3, . when taken together form the set of integers. The set of integers is usually denoted by Z. The set Z becomes practically useful, once we can combine or operate on two integers or the same integer counted twice in some or other way and then from another integer. This demands a precisely defined binary operation.

Binary Operation

Anoperation which combines two or more elements of a set to form a new element of the same set is called the binary operation.

or

Any rule which assigns to each ordered pair of elements in the set Z of integers a unique element of Z is called a binary operation on Z. In symbols, we may consider it as a function defined by

$$f\;Z\;\times\;Z\;\rightarrow\;Z$$

In practice, we use a variety of symbols such as

$$\;*\;\oplus\;,\;\otimes\;\circ\;,\;\times\;+,\;-,\;$$ to denote a binary operation.

Written in terms of elements, the definition of a binary operation looks like:

$$\;“A\;binary\;operation\;*\;on\;the\;set\;of\;integers\;Z\;associates\;each\;ordered\;pair\;of\;$$

$$integers\;(m,n)\;\in\;Z\;\times\;Z\;a\;unique\;element\;c\;\in\;Z\;$$

In such a case, the set is said to be closed under the operations of or the operation has a closure property.

Two of the most common binary operations on Z are the addition operations, denoted by ‘+’ or the multiplication operation denoted by ‘x’.

Addition: The operation of addition denoted by ‘+’says to each pair of integers m and n there is an integer x such that m + n = x.

Here, the number x is called the sum of m and n.

Multiplication: The second operation multiplication denoted by ‘x’ or ‘.’says that to each pair of integers m and n there is an integer y such that m x n = y . Here, the number y is called the product of m and n and denoted by mn, m(n), (m)n, (m)(n) or m.n .

#### Properties of Binary Operation

• Addition: Given an integer n and the number zero ‘0’, the binary operation ‘+’of addition yields n+0 = 0 + n = n Here the number 0 is called the additive identity.
• Multiplication: Given an integer n and the number ‘1’ and the binary operation ‘x’ of multiplication yields n x 1 = 1 x n =n Here, the number 1 is the multiplicative identity.
• Additive inverse (or negative): For each integer n, there is a unique integer , denoted by –n, such that n + (-n) =(-n) + n= 0. Here, the number –n is called the additive inverse.

The associative properties of Addition and Multiplication: Given three integers m,n and p

m + (n + p) = (m + n) + p and m x (n x p) = (m x n) x p

then these are that associative properties of addition and multiplication properties respectively.

Commutative properties of addition and multiplication: $$given\;the\;integers\;m,n\;\in\;z\;if\;m+n\;=n+m\;andm.n=n.m\;then\;the\;addition\;and\;multiplication\;operation\;have\;the\;commutative\;property\;respectively.\;$$

Distributive properties: Given three integers m, n and p

m x (n + p) =m x n + m x p and (m + p) x p = m x p + n x p.

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

• Any rule which assigns to each ordered pair of elements in the set Z of integers a unique element of Z is called a binary operation on Z.
• Addition: Given an integer n and the number zero ‘0’, the binary operation ‘+’of addition yields

n+0 = 0 + n = n Here the number 0 is called the additive identity.

• Multiplication: Given an integer n and the number ‘1’ and the binary operation ‘x’ of multiplication yields n x 1 = 1 x n =n Here, the number 1 is the multiplicative identity.

• Additive inverse (or negative): For each integer n, there is a unique integer , denoted by –n, such that n + (-n) =(-n) + n= 0. Here, the number –n is called the additive inverse.

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