Page 47 - DMTH401_REAL ANALYSIS

Unit 2: Algebraic Structure and Countability

 This is known as the law of Trichotomy. Then we have stated a few properties with respect Notes
to the inequality ‘’. The first property states that the inequality  is antisymmetric. The
second states the transitivity of . The third allows us to add or subtract across the inequality,
while preserving the inequality. The last property gives the situation in which the
inequality is preserved if multiplied by a positive real number, while it is reversed if
multiplied by a negative real number.
 We have also defined the bounded and unbounded intervals. The bounded intervals are
classified as open intervals, closed intervals, semi-openor semi-closed intervals. The
unbounded intervals are introduced with the help of the extended real number system
R  {– , ) using the symbols +  (called plus infinity) and – , (called minus infinity).

 There are three important aspects of the real numbers: algebraic, order and the
completeness. To describe these aspects, we have specified a number of axioms in each
case. In the algebraic aspect, an algebraic structure called the field is used. A field is a non-
empty set F having at least two distinct elements 0 and 1 together with two binary
operations + (addition) and . (multiplication) defined on F such that both + and . are
commutative, associative, 0 is the additive identity, 1 is the multiplicative identity, additive
inverse exists for each element of F, multiplicative inverse exists for each element other
than 0 and multiplication is distributive over addition. The second aspect is concerned
with the Order Structure in which, we use the axioms of the law of trichotomy, the
transitivity property, the property that preserve the inequality under addition and the
property that preserve the inequality under multiplication by a positive real number.

 In the completeness aspect, we introduce the notions of the supremum (or infimum) of a
set and state the axiom of completeness. We find that both Q and R are ordered fields but
the axioms of completeness distinguishes Q from R in the sense that Q does not satisfy the
axiom of completeness. Thus, we conclude that R is a complete-ordered Field whereas Q is
not a complete-ordered field.

 We introduce the notion of the countability of sets. A set is said to be denumerable if it is
in one-one correspondence with the set of natural numbers. Any set which is either finite
or denumerable is called a countable set. We have shown that the sets N, Z Q are countable
sets but the sets 1 (set of irrational numbers) and R are not countable.
 Thus in this unit, we have discussed the algebraic structure, the order structure and the
countability of the real numbers.

2.5 Keywords

Countable Set: A set which is equivalent to the set of natural numbers is called a denumerable
set. Any set which is either finite or denumerable, is called a Countable set.
Uncountable Set: Any set which is not countable is said to be an uncountable set.

2.6 Review Questions

1. State the properties of order relation in the set R of real numbers with respect to the
relation 3 (is greater than or equal to) and illustrate the inequality under multiplication by
a negative real number.

2. Give examples to show that the intersection of any two intervals may not be an interval.
What happens, if the two intervals are not disjoint? Justify your answer by an example.

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