Page 13 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Equivalence Relation Notes

The relation R defined on a set A is to be reflexive if aRa holds for every a belonging to A, i.e.,
(a, a) R, for every a A.
The relation R is said to be symmetric if
a R b b R a

for every ordered pair (a, b) R, i.e.,
(a, b) R (b, a) R.
The relation R is said to be transitive if
(a R b, b R c) a R c

for every a, b, c belonging to A i.e.,
[(a, b) R, (b, c) R] (a, c) R.
A relation R defined on a set is called an equivalence relation if it is reflexive, symmetric and
transitive.

Natural Numbers

The properties of natural numbers were developed in a logical manner for the first time by the
Italian mathematician G. Peano, by starting from a minimum number of simple postulates.
These simple properties, known as the Peano’s Postulates (Axioms), may be stated as follows:
Let there exist a non-empty set N such that.
Postulate I: 1 N, that is, 1 is a natural number.
+
Postulate II: For each n N there exists a unique number n N, called the successor of n.
Postulate III: For each n N, we have n 1, i.e., 1 is not the successor of any natural number.
+
+
+
Postulate IV: If m, n N, and m = n then m = n, i.e. each natural number, if it is a successor, is the
successor of a unique natural number.
Postulate V: If K is any subset of N having the properties (i) 1 K and (ii) m K m K, then
+
K = N.
The postulate V is known as the Postulate of induction or the Axiom of induction. The Principle of
mathematical induction is just based on this axiom.

Addition Composition

In the set of natural numbers N, we define addition, which shall be denoted by the symbol ‘+’ as
follows:

(i) m + 1 = m m, N
+
(ii) m + n* = (m + n)* m, n N.
The distinctive properties of the addition operation in N are the closure, associative, commutative
and cancellation laws, i.e., if m, n, p N, then
(A ) m + n N (closure law)
1
(A ) (m + n) + p = m + (n + p) (associative law)
2

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