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 $\in$ 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 $\in$ means for all the intervals discussed above:
