Page 29 - DMTH401_REAL ANALYSIS

Unit 2: Algebraic Structure and Countability

2.1 Order Relations in Real Numbers Notes

We have demonstrated that every real number can be represented as a unique point on a line
and every point on a line represents-% unique real number. This helps us to introduce the notion
of inequalities and intervals on the real line which we shall frequently use in our subsequent
discussion through out the course.
You know that a real number x is said to be positive if it lies on the right side of O, the point
which corresponds to the number 0 (zero) on the real line. We write it as x > 0. Similarly, a real
number x is negative, if it lies on the left side of O. This is written as x < 0. If x > 0, then x is a
non-negative real number. The real number x is said to be non-positive if x  0.
Let x and y be any two real numbers. Then, we say that x is greater than y if x – y > 0. We express
this by writing x > y. Similarly x is less than y if x – y < 0 and we write x < y. Also x is greater than
or equal to y (x  y) if x – y  0. Accordingly, x is less than or equal to y (x  y) if x – y  0. Given
any two real numbers x and y, exactly one of the following can hold:
either (i) x > Y
or (ii) x < y
or (iii) x = y.
This is called the law of trichotomy. The order relation  has the following properties:
Property 1
For any x, y, z in R,
(i) If x  y and y  x, then x = y,
(ii) If x  y an y  z, then x  z,
(iii) If x  y then x + z  y + z,
(iv) If x < y and o  z, then x z  y z.
The relation satisfying (i) is called anti-symmetric. It is called transitive if it satisfies (ii). The
property (iii), shows that the inequality remains unchanged under addition of a real number.
The property (iv) implies that the inequality also remains unchanged under multiplication by a
non-negative real number. However, in this case the inequality gets reversed under multiplication
by a non-positive real number. Thus, if x  y and z  0, then xz  yz. For instance, if z = –1, we see
that
–2  4  2 (–1)  4 (–1)  –2  –4.

We state the following results without proof:
 There lie an infinite number of rational numbers between any two distinct rational
numbers.
 As a matter of fact, something more is true.

 Between any two real numbers, there lie infinitely many rational (irrational) numbers.
Thus there lie an infinite number or real numbers between any two given real numbers.

2.1.1 Intervals

Now that the notion of an order has been introduced in R, we can talk of some special subsets of
R defined in terms of the order relation. Before we formally define subset, we first introduce the
notion of ‘betweenness’, which we have already used intuitively in the previous results. If 1, 2,
3 are three real numbers, then we say that 2 lies between 1 and 3. Thus, in general, if a, b and c are
any three real numbers such that a 5  b  c then we say that b lies ‘between’ a and c. Closely
related to notion of betweenness is the concept of an interval.

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