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
9. Properties of Intervals
Let § be the family of all intervals on the real line. We include in § the null set {} and single points a = [a, a]. Then the intervals have the following properties:
1. The intersection of two intervals is an interval, that is, A £ §, B £ § implies A n B £ §
2.The union of two non-disjoint intervals is an interval, that is, A £ §, B £ §,
A n B = {} implies A u B £ §
3. The difference of two non-comparable intervals is an interval, that is, A £ §, B £ §, A ¢ B, B ¢ A implies A - B £ §
Example 3.1: Let A = {2, 4), B = (3, 8). Then
A Ç B = (3, 4), A È B = [2, 8)
A – B = [2, 3], B – A = [4, 8)
2 Infinite Intervals
Sets of the form
A = {x| x > 1}
B = {x| x > 2}
C = {x| x < 3}
D = {x| x < 4}
E = {x |x £ R}