# Set Builder Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Adena Benn
Last updated: Sep 13, 2023

The set builder calculator is a helpful tool for generating and populating a set of numbers you can later use it for plotting, mathematical analysis, and much more. Keep reading this article if you want to learn:

• What intervals are and how to represent them;
• What the set builder notation is;
• How to calculate the set builder notation form for any interval;
• What is the roster form of a set;
• How to calculate the roster form from the set builder notation form; and
• Much more.

## Intervals in math: prelude to the calculations of the set builder notation form

An interval in math is a comprehensive set of numbers between two endpoints. If you imagine numbers placed in an ordered fashion on a line, an interval would be a segment on this line. By definition, intervals are defined on real numbers, so our segment is continuous.

A fairly good example of an interval is the domain of the tangent function. The tangent function is a periodic function defined as integer multiples of the interval between (excluded) $-\pi/2$ and $\pi/2$.

Let's talk endpoints. The endpoints of an interval are the starting and ending points of the segment on the number line. We can define two types of intervals, depending on the kind of endpoints:

• Closed intervals: the endpoints are included in the interval. We usually denote such interval with square brackets: $[a,b]$.
• Open intervals: the endpoints are excluded from the interval. In this case, we use curved brackets to denote the interval: $(a,b)$.
• Mixed intervals, where one of the endpoints is included, and the other is excluded. The notation is, of course, a mixture of the previous two: $[a,b)$ or $(a,b]$.

When representing an interval on the number line, we can use filled or empty dots to denote the different types of intervals and endpoints. You can see the distinction in the graphics below. Now you know how to build an interval. Let's learn how to calculate the set builder notation to restrict this interval to only specific values.

## Calculating the set builder notation

If you want to restrict an interval to a defined set of numbers following specific rules, you need the set builder notation or set builder form. To calculate it, you need:

• The interval where the set is defined;
• The set where you define the desired numbers (integer, natural, etc.); and
• Other conditions that the numbers must satisfy if specified.

Once you know all the necessary quantities, you can calculate the set builder notation for an interval by following these steps:

1. Wrap the entire expression in curly brackets as we are dealing with an enumeration of elements.

2. Begin the content of the brackets with the conventional expression $x|$, which you can translate into a set composed of elements x such that....

3. Define the interval using inequalities. It will follow these rules:

• Use the inequality signs to define the endpoints:
• An inclusive endpoint corresponds to a less/greater than and equal sign: $\leq$ or $\geq$; and

• An exclusive endpoint corresoinds to the simple inequality sign $<$ or $>$.

• Place $x$ in the center, as the number lies between the two endpoints.

The final result should look something like $a.

4. Define the set to which the numbers belong: use the set operator $\in$:

• $x\in \N$ means that $x$ belongs to the natural numbers;

• $x \in \Z$ means that the numbers belong to the integer numbers; and

• $x\in \R$ means we are dealing with real numbers.

5. Add any specified condition, for example, if you want $x$ to be odd or something similar. You can be "verbose".

🙋 Our graphing inequalities on a number line calculator may help you visualize the inequalities used in the set builder form calculator.

## Calculating the set builder form: an example

Let's say you want to isolate the odd numbers between $10$ and $23$, with the endpoints excluded. To write this set in the set builder notation, we calculate the interval first; then, we apply the conditions above.

1. The interval is $(10,23)$.

2. All real numbers in that interval correspond to the following inequality in the set builder notation we calculate:

$\{x |\ 10

3. Let's restrict the set to the integer numbers (the natural ones would have been equally good):

$\{x |\ 10

4. As we want only odd numbers, let's specify it:

$\{x |\ 10

That's it: you calculated the set builder form for this set. What if we need a complete enumeration of the elements, though? The roster form comes with our help.

## Calculating the roster form of a set

In this section, we'll learn how to calculate the roster form of a set. The roster form is an explicit enumeration of the elements of a set. We can calculate the roster form from the set builder form pretty quickly.

1. Take the interval you calculated in the set builder form and explicitly write down the members of the desired set (natural, integers...). Note that this enumeration is impossible for real and rational numbers as there are infinitely many of them between any two given (different) numbers. You can restrict them to real numbers evenly spaced by a defined quantity.
2. Apply the additional rules, removing the elements that don't follow them.

Let's apply these rules to the example above, and let's calculate the roster form from the set builder form:

1. Start with the calculated set builder form: $\{x\,|\ 10.

2. Enumerate the elements belonging to the first section ($\{x\,|\ 10): $\{11, 12,13,14,15,16,17,18,$ $19,20,21,22\}$ and notice how we dropped the endpoints.

3. As we only want odd numbers in the set, remove the even ones: $\{11,13,15,17,19,21\}$.

That's it. This is how you calculate the roster form for a set.

🙋 Our consecutive integers calculator uses a similar form to the roster form to compute specific operations. And if you are interested in bidimensional roster forms, say no more: the coordinate grid calculator applies particular rules to create sets of points in a plane.

## How to use our set builder calculator

Our set builder calculator quickly prints the possible enumeration forms of your set depending on the rules and interval you choose.

1. Start by typing the endpoints in the appropriate fields, and specify the type of endpoints (if inclusive or exclusive).

2. Choose the set to which your numbers belong. You can choose natural, integers, evenly spaced rationals, or real numbers.

3. Depending on the previous choice, you will be able to add another condition on the elements:

• If you choose integers or natural numbers, choose between all numbers, even or odd ones, or prime numbers.

• If you choose evenly spaced numbers, we'll ask you to input the distance between two numbers so that we can populate the set.

4. If possible, we will print the result in roster notation (it may be too long!), and if the calculated set builder form carries some good information, we'll publish it too.

🙋 You can input non-integer endpoints, but we will round them to the nearest integer that would belong to the set.

## FAQ

### What is the set builder notation?

The set builder notation is a mathematical tool that allows you to enumerate the elements of a set depending on the interval where they are defined and on specific rules that restrict their numbers. To calculate the set builder notation, inequalities are used to write the interval explicitly; then, other conditions are added until you reach the desired result.

### How do I represent the set builder form for the odd numbers in [5,15)?

To calculate the set builder notation for the odd numbers in [5,15), follow these easy steps:

1. Write down the interval: [5,15) corresponds to the inequality 5 ≤ x < 15.

2. Choose x such as it belongs to the natural numbers: x ∈ N.

3. Limit x to the odd numbers: x is odd.

4. Join all the previous elements to calculate the set builder notation from the interval:

{x,| 5 ≤ x < 15, x ∈ N, x is odd}

### What is the roster form of a set?

The roster form of a set is an explicit form of enumeration for the elements of a set. All elements are written down and enclosed in curly brackets in the roster form. The form is extremely helpful in catching trends and other behaviors on small-sized sets. For larger sets, the set builder notation is always preferred.

### How do you write {5,10,15,20,25} in set builder notation?

In set builder notation you can write {5,10,15,20,25} as {x,| 5 ≤ x ≤ 25, x mod 5 = 0}. In this formula, we identify two elements:

• The interval [5, 25] where all the real numbers are included; and
• The criterion that restricts the interval to the desired values (the multiples of 5).
Davide Borchia
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