REAL NUMBERS, R
One of the most important properties of the real numbers is that points on a straight line that can represent them. As in Fig 3.1, we choose a point, called the origin, to represent 0 and another point, usually to the right, to represent 1.
Then there is a natural way to pair off the points on the line and the real numbers, that is, each point will represent a unique real number and each real number will be represented by a unique point. We refer to this line as the real line. Accordingly, we can use the words point and number interchangeably.
Those numbers to the right of 0, i.e. on the same side as 1, are called the positive numbers and those numbers to the left of 0 are called the negative numbers. The number 0 itself is neither positive nor negative
10. BOUNDED AND UNBOUNDED SETS
Let A be a set of numbers, then A is called bounded set if A is the subset of a finite interval. An equivalent definition of boundedness is;
Definition 3.1: Set A is bounded if there exists a positive number M such that
|x | < = M.
for all x £ A. A set is called unbounded if it is not bounded
Notice then, that A is a subset of the finite interval [-M, M].
Example 4.1: Let A = {1, ½, 1/3…}. Then A is bounded since A is certainly a subset of the closed interval [0, 1].
Example 4.2: Let A = {2, 4, 6,…..}. Then A is an unbounded set.
Example 4.3: Let A = {7, 350, -473, 2322, 42}. Then A is bounded
Remark 3.3: If a set A is finite then, it is necessarily bounded.
If a set is infinite then it can be either bounded as in example 4.1 or unbounded as in example 4.2