Page 24 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes (ii) 1 is the multiplicative identity of N as
a.1 = 1.a = a a N.
Evidently 1 is identity of multiplication for I (set of integers), Q (set of rational numbers,
R (set of real numbers).
Inverse: An element a G is said to have its inverse with respect to certain operation o if there
exists b G such that

a o b = e = b o a.
e being the identity in G with respect to o.
–1
–1
Such an element b, usually denoted by a is called the inverse of a. Thus a o a = e = a o a for
–1
a G.
In the set of integers the inverse of an integer a with respect to ordinary addition operation is
– a and in the set of non-zero rational numbers, the inverse of a with respect to multiplication is
1/a which belongs to the set.

Algebraic Structure

A non-empty set G together with at least one binary operation defined on it is called an algebraic
structure. Thus if G is a non-empty set and ‘o’ is a binary operation on G, then (G, o) is an algebraic
structure.
(n, +), (I, +), (I, –), (R, +, .)
are all algebraic structures. Since addition and multiplication are both binary operations on the
set R of real numbers, (R, +, .) is an algebraic structure equipped with two operations.

Illustrative Examples

Example 2: If the binary operation o on Q the set of rational numbers is defined by
a o b = a + b – a b, for every a, b Q
show that Q is commutative and associative.
Solution:
(i) ‘o’ is commutative in Q because if a, b Q, then

a o b = a + b – a b = b + a – b a = b o a.
(ii) ‘o’ is associative in Q because if a, b, c Q then
a o (b o c) = a o (b + c – b c)
= a + (b + c – b c) – a (b + c – b c)

= a + b – a b + c – (a + b – a b) c
= (a o b) oc.

Example 3: Given that S = {A, B, C, D} where A = , B = {a}, and C = {a, b}. D = {a, b, c} show
that S is closed under the binary operations (union of sets) and (intersection of sets) on S.

18 LOVELY PROFESSIONAL UNIVERSITY