Open and Closed Intervals
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An interval is a continuously connected set of real numbers. For example, if we are given the two real numbers −16 and 103.5, then an interval would be all of the real numbers between −16 and 103.5. Then main question of intervals is whether the interval contains the endpoints or not. So does our interval contain −16 and 103.5 or not? So for intervals we need to distinguish between open and closed intervals. An open interval will not contain the endpoints but a closed interval will. The purpose of this lesson is to put an exact definition and picture to the idea of an open and closed interval.
Let's assume that a and b are real numbers and that <Sign in to see all the formulas>. An open interval is all of the numbers between a and b and does not include a and b. We denote the open interval between a and b as (a,b). We can use our set notation to define the open interval by
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A closed interval is all of the numbers between a and b, but does include a and b. We denote the closed interval between a and b as [a,b]. In set notation we define the closed interval by
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We can even have intervals that are half-open and half-closed. The interval
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is closed on the left side and open on the right side while the interval
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is open on the left and closed on the right. Note that the brackets and parentheses are used to denote which side is open and which side is closed. A bracket [ or ] denotes closed and a parenthesis ( or ) denotes open.
Here are all of our open and closed intervals with pictures:
| Interval | Set Notation | Picture |
|---|---|---|
| Open Interval | <Sign in to see all the formulas> |  Enlarge |
| Closed Interval | <Sign in to see all the formulas> | Enlarge |
| Half-Closed, Half-Open Interval | <Sign in to see all the formulas> | Enlarge |
| Half-Open, Half-Closed Interval | <Sign in to see all the formulas> | Enlarge |
Finally, recall that the symbol ∈ means "in a set". So <Sign in to see all the formulas> means that y (whatever it is) is in the set A. Since intervals are sets we need to show what things like <Sign in to see all the formulas> mean. <Sign in to see all the formulas> means z is a number between a and b. In other words, <Sign in to see all the formulas>. Here is what ∈ means for all the intervals discussed above:
