Subsets of a Line: Understanding Open Intervals

Open intervals are subsets of the real number line. They are typically represented using parentheses instead of brackets. The open interval from a to b contains all numbers x, where a < x < b. Consider two points a and b such that a < b. If the open interval (a, b) does not contain either point a or b, it is called an open interval. When discussing intervals, it's important to remember that endpoints are not included in the interval. Let's take a look at open intervals on the real number line.

Open Interval Notation

The notation for an open interval is (a, b), where a and b are the endpoints of the interval and a < b. So, (1, 5) represents all numbers x between 1 and 5 but does not include both 1 and 5 themselves. This is because 1 and 5 are neither less than nor greater than any other number within the interval.

Properties of Open Intervals

Some properties of open intervals are:

• Open intervals are non-empty sets. They always contain at least one element.
• If a set S is an open interval containing two points a and b with a < b, then there exists a smallest element s in S that is between a and b. Specifically, if s is the smallest element in S such that a < s < b, then s belongs to every closed interval from a to y with y > s.

Examples of Open Intervals

Here are some examples of open intervals:

• (0.75, 4.9): This includes numbers between 0.75 and 4.9, excluding these endpoints.
• (-100, -1): Numbers between -100 and -1, excluding -100 and -1.
• (3.8, 6.9): Includes numbers between 3.8 and 6.9, excluding these endpoints.

In conclusion, understanding open intervals requires familiarity with the concepts of intervals and their endpoints. Remember that open intervals are represented using parentheses and exclude their endpoints.