REAL NUMBERS, R

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}