Set Builder Calculator

Set Builder Notation Calculator

Set builder calculator is an online tool that generates a set of numbers by using interval notation. This calculator provides representations of sets in both builder and roster form.

What is the Set Builder Form?

A set builder form is an effective way to represent sets concisely and conveniently. This can be done by using a symbol or letter for an arbitrary element of the set and stating the property common to all the members. In this method, we do not list each element separately as in roster form.

For instance, the set of odd positive numbers less than 11 can be expressed in builder form as:

In roster form, this set can expressed as:

{1, 3, 5, 7, 9}

Intervals: A Foundation for Set Builder Notation

An interval is a set of all possible numbers between two fixed endpoints without any gap. It may include one or both endpoints, or neither. Interval notation is a method of representing intervals. Square brackets [ ] indicate included endpoints and parentheses ( ) indicate excluded endpoints.

How to Write Set Builder Form Using Interval?

Following are the steps to represent the set builder notation with intervals:

1. Write the opening brace

Begin with a curly brace { to enclose the set.

2. Choose a variable

Select a variable (usually x) to represent the elements of the set.

3. Specify the type of numbers (Optional)

If the set includes only specific types of numbers (e.g., integers, real numbers), indicate this using symbols such as ∈ ℤ for integers or ∈ ℝ for real numbers.

4. Insert the Vertical Bar or Colon:

Introduce either a vertical bar | or a colon : to separate the variable from the condition. This symbolizes "such that" in set builder notation.

5. Identify the interval notation

Examine the given interval and understand whether its endpoints are included or excluded. The symbols ≤ and ≥ represent included endpoints, whereas < and > denote excluded endpoints.

6. Write the closing curly brace

End with a closing brace }.

Example of Set Builder Form

Let’s consider an example to learn how to convert interval notation to set builder and roster form.

Example:

Write the set builder form from the interval (−3, 5] for the integer.

Solution:

Step 1: Choose a variable x to represent the elements of the set.

Step 2: Specify the type of numbers as integer i.e. x ∈ Z

Step 3: Insert the vertical bar | such that it separates the variable from the condition.

Step 4: Identify the interval notation. The interval excludes -3 but includes 5, hence we can write the interval notation as "-3 < x ≤ 5".

Step 5: Put everything together in curly braces. The set builder form for the given interval can be written as:

{x ∈ Z | -3 < x ≤ 5, x is integer}

Roster Form of the given interval is {-2, -1, 0, 1, 2, 3, 4, 5}

FAQs

What is the roster or Tabular form of a set?

A roster or tabular form of a set represents a set by listing its elements between curly braces and separated by commas. For instance, the set of even numbers between 1 and 10 can be expressed in roster form as {2, 4, 6, 8, 10}.

What is an evenly spaced number in a set builder?

The following form represents an evenly spaced set of numbers in set-builder notation:

{a + n. k | k ∈ Z}

Here, a is the initial term, n is the given spacing between values, and k is an integer.

For example, the set of evenly spaced numbers in the interval [8, 19) with a positive spacing of 3 can be represented as:

{x ∈ ℝ | 8 ≤ x < 19, x = 8 + 3. k, k ∈ ℤ}

This set is equivalent to the roster notation:

{8, 11, 14, 17}

How do you write the builder and roster form of an interval [-1, 3] for the real number?

The builder form of the interval [-1, 3] for real numbers would be {x ∈ ℝ | -1 ≤ x ≤ 3}. The roster form of the same interval cannot be generated as it contains an infinite number of elements.