Page 31 - DMTH401_REAL ANALYSIS

Unit 2: Algebraic Structure and Countability

Notes
Example: Test whether or not the union of any two intervals is an interval.

Solution: Let [2, 5] and [7, 12] be two intervals. Then [2, 5]  [7, 12] is not an interval as can be seen
on the line in Figure below.

However, if you take the intervals which are not disjoint, then the union is an interval. For
example, the union of [2, 5] and [3, 6] is [2, 6] which is an interval. Thus the union of any two
intervals is an interval provided the intervals are not disjoint.

2.1.2 Extended Real Numbers

The notion of the extended real number system is important since we need it in this unit as well
as in the subsequent units.

You are quite familiar with the symbols +  and – . You often call these symbols are ’plus
infinity’ and ‘minus infinity’, respectively. The symbols +  and –  are extremely useful. Note
that these are not real numbers.

Let us construct a new set R* by adjoining –  and +  to the set R and write it as
R* = R  {– , + }.
Let us extend the order structure to R* by a relation < as –  < x < + , for every x  R. Since the
symbols –  and +  do not represent any real numbers, you should, therefore, not apply any
result stated for real numbers, to the symbols +  and – . The only purpose of using these
symbols is that it becomes convenient to extend the notion of (bounded) intervals to unbounded
intervals which are as follows:

Let a and b be any two real numbers. Then we adopt the following notations:
[a, ] = {X  R: x  a}
[a, do] = {X  R: x  a}

[– , b] = {x  R: x  b}

[–, b] = {x  R: x < b}
[–,] = {X  R: –  < x < }.

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