Interval notation is a way of writing subsets of the real number line .
A closed interval is one that includes its endpoints: for example, the set { x | − 3 ≤ x ≤ 1 } .
To write this interval in interval notation, we use closed brackets [ ]:
[ − 3 , 1 ]
An open interval is one that does not include its endpoints, for example, { x | − 3 < x < 1 } .
To write this interval in interval notation, use parentheses :
( − 3 , 1 )
You can also have intervals which are half-open and half-closed:
[ − 2 , 4 )
You can also use interval notation together with the set union operator to write subsets of the number line made up of more than one interval:
[ − 4 , − 2 ] ∪ ( − 1 , 1 ) ∪ ( 1 , 2 ] ∪ { 4 }
Interval notation is a notation used to denote all of the numbers between a given set of numbers (an interval). For example, "all of the integers between 12 and 16 including 12 and 16" would include the numbers 12, 13, 14, 15, and 16. Even with such a small range of numbers, it is already cumbersome to list them. Interval notation, as well as a couple other methods, allow us to more efficiently denote intervals.
To use interval notation we need to first understand some of the commonly used symbols:
- [] - brackets denote a closed interval
- () - parenthesis denote an open interval
- ∪ - union represents the joining together of two sets
- ∩ - intersection represents the overlap between two sets
Open and closed intervals
A closed interval is an interval that includes the values on the end. The example above would be denoted as
[12, 16]
since both 12 and 16 are included. An open interval is one in which the values on the end are not included, and would be denoted as:
(12, 16)
It is also possible to have a combination of the two. If 12 were included, but 16 were not, we can denote it in interval notation as follows:
[12, 16)
The above are examples of finite intervals. It is also possible to have infinite intervals. Both negative infinity and positive infinity are considered open since it is not really possible to quantify infinity. All real numbers greater than or equal to 12 can be denoted in interval notation as:
[12, ∞)
Union and intersection
Unions and intersections are used when dealing with two or more intervals. For example, the set of all real numbers excluding 1 can be denoted using a union of two sets:
(-∞, 1) ∪ (1, ∞)
Intersection is used to denote the interval over which two sets overlap.
(-∞, 4] ∩ [2, 22]
The above reads as "the intersection between the sets (-∞, 4] and [2, 22]," which is [2, 4].
Number lines and inequalities
Intervals can also be denoted using number lines and inequalities.
Open and closed intervals
Closed intervals on number lines are denoted using filled-in circles at the endpoints; open intervals use circles that are not filled in for the endpoints:
closed interval
open interval
To express the same intervals above using inequalities:
closed interval
0 ≤ x ≤ 4
open interval
0 < x < 4
Video transcript
- [Voiceover] What I hope to do in this video is get familiar with the notion of an interval, and also think about ways that we can show an interval, or interval notation. Right over here I have a number line. Let's say I wanted to talk about the interval on the number line that goes from negative three to two. So I care about this-- Let me use a different color. Let's say I care about this interval right over here. I care about all the numbers from negative three to two. So in order to be more precise, I have to be clear. Am I including negative three and two, or am I not including negative three and two, or maybe I'm just including one of them. So if I'm including negative three and two, then I would fill them in. So this right over here, I'm filling negative three and two in, which means that negative three and two are part of this interval. And when you include the endpoints, this is called a closed interval. Closed interval. And I just showed you how I can depict it on a number line, by actually filling in the endpoints and there's multiple ways to talk about this interval mathematically. I could say that this is all of the... Let's say this number line is showing different values for x. I could say these are all of the x's that are between negative three and two. And notice, I have negative three is less than or equal to x so that's telling us that x could be equal to, that x could be equal to negative three. And then we have x is less than or equal to positive two, so that means that x could be equal to positive two, so it is a closed interval. Another way that we could depict this closed interval is we could say, okay, we're talking about the interval between, and we can use brackets because it's a closed interval, negative three and two, and once again I'm using brackets here, these brackets tell us that we include, this bracket on the left says that we include negative three, and this bracket on the right says that we include positive two in our interval. Sometimes you might see things written a little bit more math-y. You might see x is a member of the real numbers such that... And I could put these curly brackets around like this. These curly brackets say that we're talking about a set of values, and we're saying that the set of all x's that are a member of the real number, so this is just fancy math notation, it's a member of the real numbers. I'm using the Greek letter epsilon right over here. It's a member of the real numbers such that. This vertical line here means "such that," negative three is less x is less than-- negative three is less than or equal to x, is less than or equal to two. I could also write it this way. I could write x is a member of the real numbers such that x is a member, such that x is a member of this closed set, I'm including the endpoints here. So these are all different ways of denoting or depicting the same interval. Let's do some more examples here. So let's-- Let me draw a number line again. So, a number line. And now let me do-- Let me just do an open interval. An open interval just so that we clearly can see the difference. Let's say that I want to talk about the values between negative one and four. Let me use a different color. So the values between negative one and four, but I don't want to include negative one and four. So this is going to be an open interval. So I'm not going to include four, and I'm not going to include negative one. Notice I have open circles here. Over here had closed circles, the closed circles told me that I included negative three and two. Now I have open circles here, so that says that I'm not, it's all the values in between negative one and four. Negative .999999 is going to be included, but negative one is not going to be included. And 3.9999999 is going to be included, but four is not going to be included. So how would we-- What would be the notation for this? Well, here we could say x is going to be a member of the real numbers such that negative one-- I'm not going to say less than or equal to because x can't be equal to negative one, so negative one is strictly less than x, is strictly less than four. Notice not less than or equal, because I can't be equal to four, four is not included. So that's one way to say it. Another way I could write it like this. x is a member of the real numbers such that x is a member of... Now the interval is from negative one to four but I'm not gonna use these brackets. These brackets say, "Hey, let me include the endpoint," but I'm not going to include them, so I'm going to put the parentheses right over here. Parentheses. So this tells us that we're dealing with an open interval. This right over here, let me make it clear, this is an open interval. Now you're probably wondering, okay, in this case both endpoints were included, it's a closed interval. In this case both endpoints were excluded, it's an open interval. Can you have things that have one endpoint included and one point excluded, and the answer is absolutely. Let's see an example of that. I'll get another number line here. Another number line. And let's say that we want to-- Actually, let me do it the other way around. Let me write it first, and then I'll graph it. So let's say we're thinking about all of the x's that are a member of the real numbers such that let's say negative four is not included, is less than x, is less than or equal to negative one. So now negative one is included. So we're not going to include negative four. Negative four is strictly less than, not less than or equal to, so x can't be equal to negative four, open circle there. But x could be equal to negative one. It has to be less than or equal to negative one. It could be equal to negative one so I'm going to fill that in right over there. And it's everything in between. If I want to write it with this notation I could write x is a member of the real numbers such that x is a member of the interval, so it's going to go between negative four and negative one, but we're not including negative four. We have an open circle here so I'm gonna put a parentheses on that side, but we are including negative one. We are including negative one. So we put a bracket on that side. That right over there would be the notation. Now there's other things that you could do with interval notation. You could say, well hey, everything except for some values. Let me give another example. Let's get another example here. Let's say that we wanna talk about all the real numbers except for one. We want to include all of the real numbers. All of the real numbers except for one. Except for one, so we're gonna exclude one right over here, open circle, but it can be any other real number. So how would we denote this? Well, we could write x is a member of the real numbers such that x does not equal one. So here I'm saying x can be a member of the real numbers but x cannot be equal to one. It can be anything else, but it cannot be equal to one. And there's other ways of denoting this exact same interval. You could say x is a member of the real numbers such that x is less than one, or x is greater than one. So you could write it just like that. Or you could do something interesting. This is the one that I would use, this is the shortest and it makes it very clear. You say hey, everything except for one. But you could even do something fancy, like you could say x is a member of the real numbers such that x is a member of the set going from negative infinity to one, not including one, or x is a member of the set going from-- or a member of the interval going from one, not including one, all the way to positive, all the way to positive infinity. And when we're talking about negative infinity or positive infinity, you always put a parentheses. And the view there is you could never include everything all the way up to infinity. It needs to be at least open at that endpoint because infinity just keeps going on and on. So you always want to put a parentheses if you're talking about infinity or negative infinity. It's not really an endpoint, it keeps going on and on forever. So you use the notation for open interval, at least at that end, and notice we're not including, we're not including one either, so if x is a member of this interval or that interval, it essentially could be anything other than one. But this would have been the simplest notation to describe that.