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Set-Builder Notation

Sometimes, we use roster form to describe sets. When we use roster form (or roster method), we simply list, or enumerate, each element of the set within a set of curly braces. For example, the following set is in roster form: $$\{0, 1, 2, 3, 4, 5 \}$$

But what if we had a really big set—one with a large number of elements that all share a certain property? For example, what if we wanted to describe the set of integers from $0$ to $100,$ inclusive? That's a lot of numbers, and it would take a while to write all of them out.

In this scenario, we might want to use set-builder notation. This means that instead of listing every single element of the set, we define the set based on conditions, (also called rules, properties, or predicates). So, if all members of your set meet a certain condition, set-builder notation may be the way to go.

Here's how it's set up:

Let's say our variable is $x.$ In set-builder notation, we would write: $$\{x: \text{conditions}\}$$ or $$\{x \:| \text{ conditions}\}$$

This is the set of elements $x$ such that $x$ satisfies certain specified conditions. We will see examples of such conditions next.

Important note: We read the colon, $:,$ or the bar, $|,$ as "such that."

Let's revisit our set of integers from $0$ to $100.$ All elements of this set meet the conditions that they are integers and they are between $0$ and $100,$ inclusive. Thus, instead of describing this set in roster form, we can use set-builder notation like so: $$\{x : x \in \mathbb{Z} \text{ and }0 \leq x \leq 100\}$$

This reads as "$x$ such that $x$ is an integer and $x$ is between $0$ and $100,$ inclusive." Notice that this is more concise than writing out all the integers from $0$ and $100$ in the roster form of the set. However, sometimes it is easier to express a set in roster form, depending on the situation.

Expressing a Set using Set-Builder Notation

Right! This set is in roster form because each element is listed separately. However, we can also describe this set with set-builder notation.

The first thing we need to do is find conditions met by each element of the set. First, notice that all elements of $S$ are between $5$ and $9,$ inclusive. Next notice that all of the elements of $S$ are integers. Now that we know the two necessary conditions elements in $S$ satisfy, we can write the set in set-builder notation.

First we write our curly brackets: $$\{ \phantom{00000000} \}$$

Next we write our variable, in this case let's say $x:$ $$\{x\phantom{0000000}\}$$

Next we write a bar or a colon to separate the variable from the conditions. In this case, let's use a bar. Remember, the bar means "such that." $$\{x\: | \phantom{000000}\}$$

Finally, we include our two conditions on the right of the bar. $$\{x\: | \: x \in \mathbb{Z} \text{ and } 5 \leq x \leq 9\}$$

Many Ways to Express the Same Set

Let's do another example! This one is a little tricky.

As it turns out, there may be more than one way to describe a set using set-builder notation! Let's take a closer look at our set: $$X = \{3, 5, 7\}$$ Notice that $3, 5,$ and $7$ are all integers, so we will want to include that in our list of conditions. Also notice that each element of $X$ is between $3$ and $7,$ inclusive, so we will also want to include that as a condition.

However, not all integers from $3$ to $7$ are included in this set. For example, $4$ is not in $X$. So, we need another condition in our set-builder notation to make it clear that $X$ only consists of $3, 5,$ and $7.$

What do $3, 5,$ and $7$ all have in common? Well for one, they're all odd. But, if you think about it a little more, you can also see that they're all prime. $3, 5,$ and $7$ are the only odd numbers between $3$ and $7,$ inclusive, and they are also the only prime numbers between $3$ and $7,$ inclusive. Thus, our third condition in our set-builder notation will be either that the elements of $X$ are odd or the elements of $X$ are prime. (Saying both would be redundant.)

So, our set-builder notation for $X$ might look like either of these, if we want to be more wordy:

• $X=\{x : x \text{ is an odd integer between } 3 \text{ and } 7, \text{ inclusive}\}$
• $X=\{x : x \text{ is a prime integer between } 3 \text{ and } 7, \text{ inclusive}\}$

Or, it might look like either of these if we want to use more mathematical notation:

• $\{x : x \in \mathbb{Z}, \:x \text{ is odd}, \:3 \leq x \leq 7\}$
• $\{x : x \in \mathbb{Z}, \:x \text{ is prime}, \:3 \leq x \leq 7\}$

We are trying to describe the set of numbers between $0$ and $2,$ non-inclusive, so one of our conditions will be $0 < x < 2.$ Let's find another condition. We know we are including all numbers between $0$ and $2,$ not just integers. For example, we can include $1.5$ in our set, even though it is not an integer. Thus, our other condition will be that $x$ is a real number, or $x \in \mathbb{R}.$

Putting everything together, we now can express the set using the following set-builder notation: $$\{x : x \in \mathbb{R}, \: 0 < x < 2 \}$$

Extension of Set-Builder Notation

Sometimes, instead of a single variable before the condition, you will see a term that more closely resembles a function. For example, instead of $$\{x : x \geq 0, \sqrt{x} \in \mathbb{Z}\} = \{0, 1, 4, 9, 16, 25, \dots\}$$ you might see something like $$\{x^2 : x \in \mathbb{Z}\} = \{0, 1, 4, 9, 16, 25, \dots\}$$ or the set of all perfect squares.

How do we know that $\{x^2 : x \in \mathbb{Z}\}$ is the set of perfect squares? First, we look at the conditions. Here, we can see that $x \in \mathbb{Z},$ or $x$ is an integer. Then, we look at the term, or the function to the left of the condition. This represents the elements in our set. Here, our term is $x^2,$ so our set consists of squares of integers.

Let's let $x$ equal some integers, and then see what happens when we square them. $$\begin{align} &:\\ &:\\ &:\\ x=-2 &\rightarrow x^2=4\\ x=-1 &\rightarrow x^2=1\\ x=0 &\rightarrow x^2=0\\ x=1 &\rightarrow x^2=1\\ x=2 &\rightarrow x^2=4\\ x=3 &\rightarrow x^2=9\\ x=4 &\rightarrow x^2=16\\ &:\\ &:\\ &:\\ \end{align}$$ Remember, if we square a negative integer, we will get the same result as if we square its positive counterpart. Thus, the set of squares of integers looks like this: $$\{0, 1, 4, 9, 16, \dots\}$$

Basically, with extended set-builder notation, you have to do an extra step and perform a function on your variable to find the elements of the set.

Instead of a single variable $x$ in the left of the colon, we have a function $x+1.$ Since we are dealing with extended set-builder notation, first look at the conditions. To find the elements of the set, we will perform the function on integers between $2$ and $5,$ non-inclusive.

What integers are between $2$ and $5?$ $3$ and $4,$ of course! Now, all we have to is plug $x=3$ and $x=4$ into our $x+1$ function to find the elements of the set. $$\begin{align} x=3 \rightarrow x+1=4\\ x=4 \rightarrow x+1=5\\ \end{align}$$

So, the elements of our set are $4$ and $5,$ and out set in roster form is $\{4, 5\}.$

Wrap-Up

The most important things to take away from this explanation are

1. Set-builder notation looks like this: $$\{\text{variable} : \text{conditions}\}\\ \text{or}\\ \{\text{variable} \: | \:\text{conditions}\}$$
2. Set-builder notation is helpful when you want to describe elements of the set based on a rule or a condition, rather than listing each individual element.
3. There may be more than one way to express a set using set-builder notation or extended set-builder notation.

Builder Notation

Set-builder notation is the mathematical notation for describing a set by stating all the properties that the elements in the set must satisfy.

The set is written in this form: {variable $\mid$ condition1, condition2,...}. The bar in the middle can be read as "such that".

Builder notation often uses math specific symbols such as $\in$, $N$, or $Z$. These symbols have special meanings in math, and if you see a new one you can always look it up. Here's a short list of ones you might see commonly:

• $\in$ means "is in". So if you see $x \in N$, that means that $x$ can only be numbers that are part of $N$.
• $N$ or $\mathbb{N}$ means "the natural numbers", or all positive integers.
• $Z$ or $\mathbb{Z}$ means "integers".
• $Q$ or $\mathbb{Q}$ means "rational numbers", or any number that can be expressed by a fraction.
• $R$ or $\mathbb{R}$ means "real numbers", or any number that isn't imaginary. If you don't know what that means, don't worry! It just means all the numbers.

Examples:

$$\{x | x \in N, x \le 5\}$$

$x \in N$ means that $x$ is in the natural numbers, so $x$ has to be an integer above $0$. $x \le 5$ means that $x$ has to be equal to or less than $5$. So, we could rewrite this set in roster notation as $\{1, 2, 3, 4, 5\}$.

$$\{x | n \in N, x = 2n\}$$

Again, we know $n$ has to be a positive integer. $x$ is the set of numbers we want, and $x$ is equal to $2$ times all possible values of $n$, so $x$ must always be even. This means that our set is all positive even integers.

$$\{(a, b) | \frac{a}{b} \in N\}$$

For this expression, we're looking for a set of pairs. The pairs will always be related to each other in a consistent way—in this case, $a$ is always a numerator of a fraction that $b$ is the denominator for. This fraction always has to fit into the natural numbers, so it has to be a fraction that can be simplified into a positive integer.

For example, $(6, 3)$ is a member of this set, since it would form the fraction $\frac{6}{3} = 2$, which is a natural number. However, $(3, 6)$ is not a member of the set, since $\frac{3}{6} = 0.5$, which is not an integer and thus not a natural number.

If we look at the set in roster notation, we can notice that it's all the multiples of $3$ except $0$. These are all integers, but not all positive numbers, which means they're part of $Z$. Multiples of $3$ are also always divisible by $3$, so we can just say that the set is equal to all $x$ divisible by $3$ that are still integers, or $\frac{x}{3} \in Z$.

But wait, that doesn't quite work--$0$ is an integer, and it's technically divisible by $3$! So we need to include another term that says that $x$ cannot be $0$: $x \ne 0$.

So, finally, we get: $$\{x | \frac{x}{3} \in Z, x \neq 0\}$$

Writing sets using basic bracket notation is a fine way to describe small sets with only a few elements, as in $\{1, 3, 5, 7\}$ or $\{\text{Barack Obama}, \text{Joe Biden}\}$. However, this way of writing a set is less useful when we get more adventurous with the sets we are describing:

• How would we describe the set which contains all of the odd counting numbers?
• What about the set of people who work for President Barack Obama?

In these situations, either the set of elements we want to describe is very large (even infinitely large!), or it is described in terms of a property of the elements, rather than their identities. This is where set builder notation comes in:

• The set of odd counting numbers can be written as: $\{\text{Integers } n \mid n > 0 \text{ and } n \text{ is odd}\}$ (read as "the set of integers $n$ such that $n$ is greater than $0$ and $n$ is odd")
• The set of people working for Barack Obama: $\{\text{People } p \mid p \text{ works for Barack Obama}\}$

How it Works

In general, here's how it works. To describe a set using set builder notation, you start with a larger set, called the universe, and then you form a new set by describing the elements you want to pick out from the universe. More specifically, we give a name for a general element of the universe, and then use a logical formula to describe which elements are contained in the set.

To check if an element of the universe is contained in our set, we just plug that element into our formula, and see if it satisfies the whole logical statement.

For the first example above (describing the odd counting numbers):

• The universe for our set builder notation is "the integers"
• We named our general integer "$n$"
• Our formula involving the general element $n$ is "$n > 0 \text{ and } n \text{ is odd}$"

To check if an integer is contained in this set, we plug it into the formula in place of $n$. For instance, if we wanted to ask whether $-3$ was in this set, we'd check whether "$-3 > 0 \text{ and } -3 \text{ is odd}$". Since $-3$ is not greater than $0$, we know that it isn't in our set. However, $17$ is in our set because "$17 > 0 \text{ and } 17 \text{ is odd}$"

Your Turn!

Now it's your turn! Let's see if you really understand the different parts of set builder notation. In the second example above, about Barack Obama, can you identify the components of the set builder notation we used to describe a set?

Set builder notation is the conventional format in which mathematicians define a set of elements. We use this notation to simply what we need to write as larger sets can be more complicated.

The general format of a set in set builder notation is as such: $$A=\{\text{possible elements } | \text{ condition(s)}\}$$

Lets break it down. We first define our set in curly brackets. In the first section, we need to indicate what type of elements we want in our set, the vertical line indicates "such that", and the statement(s) after the line will be our conditions. Make sure to remember though that this notation for "such that" can only be used when defining sets!

So what do we mean by the types of elements we want in our set? For example, we might want to only work with integers and don't want fractions or decimals, then we would use numbers in $\mathbb{Z}$. We call this the universal set for the set you are constructing. Here is a list of some common examples of universal sets that are commonly used:

• Natural Numbers($\mathbb{N}$): All integers greater than or equal to $1$
• Integers($\mathbb{Z}$): Negative whole numbers, positive whole numbers, and $0$
• Rational Numbers($\mathbb{Q}$): All numbers that can be expressed as an integer divided by another integer
• Real Numbers($\mathbb{R}$): All numbers that don't don't include imaginary numbers
• Complex Numbers($\mathbb{C}$): All real numbers and imaginary numbers combined

If you want to learn more about these common sets, you can watch this video:

(Video) Number Sets

by SmithMathAcademy

This video by SmithMathAcademy describes number sets.

The second part of the set builder notation is the conditions. Now that we know what types of elements we want, for example integers, we can find the specific elements we want within the universal set. We indicate this with a single or multiple mathematical statements, depending on how complex you want your set to be. Below are some examples.

If we want to have a set of all numbers from $1$ to $5$: $$A = \{x\in \mathbb{N} | x\leq 5\}$$

If we want all integers between $-2$ and $2$: $$B = \{y\in \mathbb{Z}|x>-2\text{ and }x<2\}$$

Make sure that you always indicate the universal set as this gives readers the context of the set so they know what you mean specifically.

It is also possible to make sets with elements that are sets or pairs of numbers, such as coordinates.

Here is an example for defining a set of pairs of numbers: $$C=\{(x,y)\in \mathbb{Z}^2 | y=x^2\}$$

This consists of all integer pairs of $(x, y)$ that satisfy the equation $y=x^2$. $(x,y)\in \mathbb{Z}^2$ denotes that both $x$ and $y$ are part of $(x,y)\in \mathbb{Z}$.

(Video) An Introduction to Set Builder Notation, Intermediate Algebra, Lesson 28

by Learn Math Tutorials

This video by Learn Math Tutorials covers set-builder notation.

$\large{\text{Summary}}$

Set-builder notation has two parts: the variable, and the conditions. $$\{\text{variable}: \text{conditions}\}$$

This denotes the set of elements $(x)$ that satisfies all of the conditions stated in the set.

Example $1$:

$$\{x|x>2\}$$

This means all values that are greater than two. In the video, this is demonstrated on the number line. At the value of $2$ there is a parenthesis because we are not included $2$ in the set.

Example $2$:

$$\{x|x\le{-1}\}$$

This means all the values that are less than or equal to negative one. On the number lines, all the values to the left of $-1$ are shaded and we use a square bracket.