Page 23 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Types of Binary Operations Notes

Binary operations have the following types:
1. Commutative Operation: A binary operation o over a set G is said to be commutative, if
for every pair of elements a, b G,

a o b = b o a
Thus addition and multiplication are commutative binary operations for natural numbers
whereas subtraction and division are not commutative because, for a – b = b – a and a b =
b a cannot be true for every pair of natural numbers a and b.
For example 5 – 4 4 – 5 and 5 4 = 4 5.
2. Associative Operation: A binary operation o on a set G is called associative if a o (b o c) =
(a o b) o c for all a, b, c G.
Evidently ordinary addition and multiplication are associative binary operations on the
set of natural numbers, integers, rational numbers and real numbers. However, if we
define a o b = a – 2b , a, b R
then (a o b) oc = (a o b) – 2c = (a – 2b) – 2c = a – 2b – 2c

and a o (b o c) = a – 2(b o c) = a – 2(b – 2c)
= a – 2b + 4c.
Thus the operation defined as above is not associative.
3. Distributive Operation: Let o and o be two binary operations defined on a set, G. Then the
operation o is said to be left distributive with respect to operation o if
a o (b o c) = (a o b) o (a o c) for all a, b, c G
and is said to be right distributive with respect to o if,

(b o c) o a = (b o a) o (c o c) for a, b, c, G.
Whenever the operation o is left as well as right distributive, we simply say that o is
distributive with respect to o.

Identity and Inverse

Identity: A composition o in a set G is said to admit of an identity if these exists an element
e G such that
a o e = a = e o a a G.
Moreover, the element e, if it exists is called an identity element and the algebraic structure
(G, o) is said to have an identity element with respect to o.

Examples:
(i) If a R, the set of real numbers then 0 (zero) is an additive identity of R because
a + 0 = a = 0 + a a R
N the set of natural numbers, has no identity element with respect to addition because
0 N.

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