1.7: Intervals and Interval Notation
- Page ID
- 174162
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| Title | Topics | Type | Length |
|---|---|---|---|
| Review of Interval Notation |
|
Review | 15:05 |
| Another Review of Interval Notation |
|
Review | 13:30 |
An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset.
Examples are the numbers \(x\) from one to two, \( 1 \leq x \leq 2\), and the numbers \(y\) greater than \(10\), i.e. \(y \gt 10\). In particular, the empty set \(\varnothing\) and the entire set of real numbers \(\mathbb{R}\) are both intervals.
The endpoints of an interval are its supremum (least upper bound), and its infimum (greatest lower bound), if they exist as real numbers.
If the infimum does not exist and the interval is not empty, one says often that the corresponding endpoint is negative infinity, written \(-\infty\). Similarly, if the supremum of a non-empty interval does not exist, one says that the corresponding endpoint is positive infinity, written \( \infty \). Non-empty intervals are completely determined by their endpoints and whether each endpoint belongs to the interval.
Open and Closed Intervals
An open interval does not include any endpoint.
An open interval can be succinctly indicated with parentheses.
The following are examples of open intervals.
- The interval of all real numbers greater than \(0\) and less than \(1\) is written \((0,1)\).
- The open interval \( (0,+\infty)\) consists of real numbers greater than \(0\), i.e., positive real numbers.
- \(\{ x \in \mathbb{R} \mid a \lt x \lt b \} = (a,b)\)
- The set of all real numbers is \((-\infty, +\infty) \)
- \((a,a) = \varnothing\)
A closed interval is an interval that includes all its finite endpoints.
When both endpoints are finite, they are enclosed in square brackets.
The following are examples of closed intervals.
- \([0,1]\) is the closed interval with contents greater than or equal to \(0\) and less than or equal to \(1\).
- \(\{ x \in \mathbb{R} \mid a \leq x \leq b \} = [a,b]\)
- The set of all real numbers less than or equal to \(-2\) is \((-\infty, -2]\).
Half-Open Intervals
A half-open interval has two distinct finite endpoints, and includes one but not the other. It is said to be left-open or right-open depending on whether the excluded endpoint is on the left or on the right.
These intervals are denoted by mixing notations for open and closed intervals.
The following are examples of half-open intervals.
- \([0,1)\) is the half-open interval with contents greater than or equal to \(0\) and strictly less than \(1\).
- The set of real numbers greater than \(a\) but at most \(b\) is \((a,b]\).
In summary, a set of the real numbers is an interval if and only if it is an open interval, a closed interval, or a half-open interval.
Unions and Interval Notation
We have encountered unions before (when reviewing sets). When it comes to joining all elements from two intervals, unions can be very useful; however, as the following example shows, care must be taken to not overstate an interval.
- The set of all real numbers less than \(-2\) or greater than or equal to \(7\) is \((-\infty,-2) \cup [7,\infty)\).
- \( \{ x \in \mathbb{R} \mid x \gt 5 \text{ or } x \leq 20 \} = (-\infty,\infty)\)
- \( [2, \infty)\) is the set of all numbers greater than \(5\) or greater than or equal to \(2\).
Part \(b\) of this last example is often confusing to students. Remember, we want all numbers greater than \(5\), but then we also want all numbers less than or equal to \(20\). Take a moment to try to find a number that is not in one of these sets (hint... you won't).
Unions of intervals come up often in Algebra when solving equations. You might have an equation where solutions could be from the bucket of numbers less than \(-2\) or from the bucket of numbers greater than or equal to \(7\). In this case, you would write your solution set as \((-\infty,-2) \cup [7,\infty)\).
Intersections and Interval Notation
Intersections are more commonly found in Algebra and Calculus when determining domains of functions. The key difference in language between a union and an intersection is the conjunction. In unions, you're willing to take values from one set or the other. In intersections, you want values to be in both sets. That is, you want numbers that are both in one set and the other set.
- The set of all real numbers less than \(-2\) and greater than or equal to \(7\) is \(\varnothing\).
- \( \{ x \in \mathbb{R} \mid x \gt 5 \text{ and } x \leq 20 \} = (5,20]\)
Additional Examples
Use interval notation to indicate all real numbers greater than or equal to \(−2\).
- Solution
-
Use a bracket on the left of \(−2\) and parentheses after infinity: \([−2,\infty)\). The bracket indicates that \(−2\) is included in the set with all real numbers greater than \(−2\) to infinity.
Use interval notation to indicate all real numbers between and including \(−3\) and \(5\).
- Answer
-
\([−3,5]\)
Write the interval expressing all real numbers less than or equal to \(−1\) or greater than or equal to \(1\).
- Solution
-
We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to \(1\). So, this interval begins at \(−\infty\) and ends at \(−1\), which is written as \((−\infty,−1]\).
The second interval must show all real numbers greater than or equal to \(1\), which is written as \([1,\infty)\). However, we want to combine these two sets. We accomplish this by inserting the union symbol, \cup , between the two intervals.
\[(−\infty,−1]\cup[1,\infty) \nonumber\]
Express all real numbers less than \(−2\) or greater than or equal to \(3\) in interval notation.
- Answer
-
\((−\infty,−2)\cup[3,\infty)\)
Sources
Several parts of this text use modifications from the following source:
- Wikipedia article: "Interval (Mathematics)"
This source is released under the Creative Commons Attribution-Share-Alike License 4.0.