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Set Notation and Interval Notation

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Today we're going to talk about set notation and interval notation. Set notation and interval notation are ways to represent a range of values within certain limitations. So we're going to talk about the format for writing within each notation. And then we'll do some examples.

So let's start by talking about set notation. I've got a graphical representation of a set of numbers. And I'm going to show you how you would write that using set notation.

So I can see that my set ranges from negative 2 to positive 2. And so we use the open and the closed circles to show the set does include negative 2, so a filled-in circle, and it does not include the exact value of positive 2, so an open circle. But another way to represent whether a value is included or excluded in a set is by using a curved bracket to show that it is not included in the set and a square bracket to show that the exact value is included in the set. So you would use these brackets right on the number line instead of using a filled-in circle or an open circle.

So this represents the set of values from negative 2 to positive 2 that is in our set, again including negative 2, but not including the exact value of positive 2. So if we want to write this in set notation, we would write it as all x values such that x is going to be greater than or equal to-- so x is greater than or equal to negative 2, but less than positive 2. So this is how we would write this in set notation.

All right. Let's talk about interval notation. So again I have a graphical representation of a set of numbers. And I'm going to show how you represent that set of numbers using interval notation.

So we start by noticing that our set ranges from negative 1 to 4. And I can see that the open circle is indicating that it does not include the exact value of negative 1. But it does include the exact value of 4. And so again, we can represent those on our number line with a square bracket, for it does include the value, and a curved bracket bracket, which indicates it does not include the exact value. And we actually use these curved brackets and square brackets for our interval notation.

So the way I would write this in interval notation is I write my smallest value first, so negative 1. And I write my largest value second, so positive 4, separated by a comma. And then I use my brackets to indicate whether it includes or does not include that number. So a square bracket with the 4, because again it does include 4, and a curvy bracket with the negative 1, because it does not include the exact value of negative 1. So this is how you would write this set in interval notation.

So here I've got a graph representing a set of numbers or an inequality on a number line. I see that the lowest point of my set is at negative 4 and then it continues and includes all numbers that are greater than negative 4, but because of the open circle, not including negative 4. So I'm going to write this in set notation.

So I know that my set notation starts with our bracket. We have x and the vertical line. And I want to show that x is greater than negative 4 but not equal to negative 4. So I'll have x is greater than negative 4. So the way that you read this in set notation is "all x values such that x is greater than negative 4."

All right. For our second example, we again have a graph of an inequality or set of numbers on a number line. Let's write this in interval notation. So I see that my set starts at 1 and then goes to numbers that are less than negative 1, or sorry, less than positive 1. And we can also see that it does include the exact value of 1 because of the closed circle.

So when I'm writing this in interval notation, my smallest value actually is going to be negative infinity, because it's going on towards infinity in the negative direction. So when I write it in interval notation, I'm going to start with negative infinity as my smallest value, separated by a comma, and then my largest value that the set includes, which is 1.

So I know I'm going to use a square bracket at 1, because it includes the exact value of 1. And I'm going to use a curved bracket or a parentheses for negative infinity. You always use a parentheses or curved bracket when you're using positive or negative infinity in interval notation, because infinity has no bound. You cannot have an exact value of infinity, so you always use a parentheses with either positive or negative infinity when you're writing in interval notation.

Let's do one more example. So for this example, I've got a number line with two highlighted regions or ranges that x can be. So first I have that x can be less than or equal to negative 1, or x can be greater than or equal to positive 1. So I'm going to show you how to write that first in set notation and then interval notation.

So our set notation starts like this. And I first want to write that x is less than or equal to negative 1, so x is less than or equal to negative 1. Again because of our filled-in circle, we know that it's less than or equal to.

Or x can be greater than or equal to positive 1. Same thing here, filled-in circle, so x can be exactly equal to positive 1. So this is how you would represent these two ranges using set notation.

In interval notation, we want to again represent both ranges. So if I start with this range, I know that it starts at negative infinity, and then is everything up to and including negative 1. So my smallest value in my interval is negative 1. I separate it with a comma. And my largest value in this first interval is negative 1.

I am going to use a square bracket at my negative 1, because it includes negative 1. And I always use a parentheses or curved bracket with either positive or negative infinity.

I also want to represent this range with my interval notation, so I'm going to start with a square bracket, because 1 is my smallest value in this set. And it can be exactly 1, so I have a square bracket, and then separated by a comma. And then my largest value, because it goes all the way up to positive infinity. So again with infinity, just as with negative infinity, we always use a parentheses or curvy bracket.

Now, I want to show that I am including both of these sets, so we use a U symbol with interval notation to represent the union of both of our individual sets. So this is how you would write our two sets using interval notation.

So here's a summary of the different ways that we can write inequalities using set notation and interval notation. Make sure you get these in your notes as it will be useful for you as you're doing more examples.

So I hope that these examples and notes helped you understand a little bit more about set in interval notation. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.