Set Builder Notation – Symbols, Forms of Writing Sets & Examples

When we write a set, we usually list all the elements inside curly brackets. For example, the set {1, 2, 3} includes the numbers 1, 2, and 3. But what if the set has too many elements, or even an infinite number, like all the numbers between 2 and 10? In such cases, listing them is not practical.

That’s where set builder notation helps. Instead of listing the elements, we describe them using a rule or condition. This notation tells us what kind of elements belong in the set based on certain properties they must follow.

We write it like this:
X = {x | condition on x}
or
X = {x : condition on x}

This is read as: “X is the set of all x such that x meets the given condition.”

For example, the set of all real numbers greater than 2 and less than 10 is written as:
A = {x ∈ ℝ | 2 < x < 10}

This means that A contains all real numbers between 2 and 10 (but not including 2 and 10 themselves).
In this maths article, we will learn about sets, set-builder notation definition, various symbols involved, how to read these notations followed by the various uses, solved examples and FAQs.

What are Sets?

Sets are collections of distinct elements grouped together based on a shared characteristic or property. The elements can be numbers, letters, symbols, or even other sets. Each element within a set is considered unique, and the order of the elements is generally not important. Set builder notation is one of the ways of representing sets in maths. They can be finite or infinite. 

Sets can be finite, meaning they have a specific number of elements, or infinite, meaning they continue indefinitely. For example, the set of all positive even numbers {2, 4, 6, ...} is infinite. Sets can also be empty, denoted by {}, when they have no elements. This is called the empty set or the null set. We can also perform operations on sets such as union, intersection, and complement which help us to compare, combine, and analyse many mathematical objects.

Methods to Represent a Set

The different ways of writing sets in maths are as follows:

1. Listing Method or Roster Method
2. Rule Method or Set Builder Form

Sets in Listing Method or Roster Method

In the listing method or roster method, all the elements of a given set are placed within curly brackets with the elements separated by commas and written once.

Example of listing method or roster method: the set of letters in the word, “WordPress” is written as X = {W, o, r, d, P, e, s}.

Sets in Rule Method or Set Builder Form

In set builder form or rule method, the elements are represented with some relation between them. Know how to make set builder notation with the below example.

Example of rule method or set builder form: For a given set P with elements {2, 3, 5, 7, 11, 13}

This can be written as:

P= {x: x is a prime number less than 17}

or

P= {x : x prime number<17}

or

P= {x | x prime number<17}

This is read as P includes elements x such that x is a prime number that is less than “17”.

Learn about Types of Sets and Union of Sets

What is Set Builder Notation?

In set theory, the different notations or patterns applied for specifying the sets are tabular form, set builder, and simple descriptive form.

In set builder notation we represent the elements of a set X in the form X = {x | (conditions followed by x or properties of x)} OR X= {x : (condition followed by x or properties of x)}. This is read as X is the set of all elements x such that they all satisfy (condition of x or properties of x). 

For example we can represent the set of all integers greater than zero in roster form as {1, 2, 3,...} whereas in set builder form the same set is represented as {x: x ∈ Z, x>0} where Z is the set of all integers.

As we can see the set builder notation uses symbols for describing sets. These notations are employed to define the different elements of a given set, relationships between them and the different operations among the sets.

Set Builder Notation Symbols

Different symbols involved in representing the set builder notation are given below:

Symbol	Meaning
W	Whole Numbers(W) are numbers that start with zero.
N	Natural Numbers (N); these are numbers that start from 1.
Z	Integers (Z) include all positive and negative numbers.
Q	Rational Numbers (Q) are expressed in the form of a/b.
R	Real numbers (R) include whole numbers, rational numbers and irrational numbers.

Other related symbols used in set builder notation are:

Symbol	Meaning
{ }	Sets are represented using these brackets.
| or :	The symbol means “such that”.
∈	The symbol denotes “is a component or element of”.
∉	The symbol denotes “is not a component or element of”.

How to Read Set Builder Notation

Set builder notations are sorts of notations used for expressing a set by exhibiting properties that all its components must satisfy. Now let us learn how to read these notations with the format and example:

The basic format that is applied for such notations is:

P= { x | condition regarding x }

or

P = { x : condition regarding x }

You can understand this as in this approach of representing a set, we document the element using a variable followed by a vertical line/colon and compose the available property for the representative element.

Likewise in the below example, you can state that; set Q is read as having ‘a’ number of elements such that ‘a’ is an odd number that is less than 12. Also, you should notice that the colon can be substituted by a vertical bar or vice versa as these are read in the same manner.

Q = { a | a is an odd number, a<12 }

Learn about Cartesian Product of Sets and Subsets

Uses of Set Builder Notation

The knowledge to reason why or the use of set builder notation is equally important to understand the concepts and definitions. Consider a situation where you have to list all the natural numbers between 5 to 12, this can be simply done through the listing method.

Now suppose you have to list a huge data set like the set of integers from 0 to 450 or say a set of all real numbers between 5 to 15, etc. Such notations are difficult to draft in the listing method format as a lot of numbers are involved in such sets. Here comes the use of set builder notation.

As in set builder notation instead of recording all the elements of the set one by one, you can specify the sitting arrangement of the elements based on conditions that would include rules or properties. The condition is formulated in such a way that all the components of the sets follow the same.

In a reverse manner, you can also understand this as if all components of the set satisfy a specific condition then set builder notation can be the preferred approach.

Learn about Cantor Set

Set Builder Notation and Interval Notation

In an interval notation representation, simply the start and end numbers of the interval are used for the representation. That is, in this type of notation only the lower limit and upper limit points are mentioned.

When the endpoints are framed in the [ ] bracket then it says that the end values are included.

And when endpoints are framed in the ( ) bracket then it says that the end values are not included.

For example, if the notation is as shown; (7,28].

Then you can say that the particular interval notation of the set includes all real digits between 7 and 28, where 28 is contained in the set while 7 is not a component of the set.

Now let us understand how we can relate the set builder notation with the interval notation. Imagine you are given the following examples:

{x |-12 < x < 51}

This can be represented in interval notation as; (-12, 51).

Learn about various Set Operations and Cantors Diagonal Argument

The domain of a function represents the set of all possible input values or x-values for which the function is defined. It can be described using set builder notation as follows:

Domain: {x | condition on x}

Here, the vertical bar "|" is read as "such that" and the condition on x specifies any restrictions or criteria that the input values must satisfy.

Example of Domain in set builder notation:

Consider the function f(x) = sqrt(x).

The domain of this function can be expressed as:

Domain: {x | x is a real number, x ≥ 0}

This notation indicates that the domain of the function h(x) is the set of all real numbers greater than or equal to 0.

Similarly, the range of a function represents the set of all possible output values or y-values that the function can produce. It can be described using set builder notation as follows:

Range: {y | condition on y}

Again, the vertical bar "|" is read as "such that" and the condition on y specifies any restrictions or criteria that the output values must satisfy.

Example of Range in set builder notation:

Consider a function y = sin(x).

The range of this function can be expressed as:

Range: {y | y is a real number, -1 ≤ y ≤ 1}

This notation indicates that the range of the function y is the set of all real numbers between -1 and 1.

Properties of Set Builder Notation

1. Describes a Set Using a Rule
   • Set builder notation defines the elements of a set by a condition or property that they satisfy.
   • Example: {x | x > 0} means all x such that x is greater than 0.
2. Compact Representation
   • It helps to write sets with many or infinite elements in a short and neat way.
   • Example: Instead of listing all even numbers, we write: {x | x = 2n, n ∈ ℤ}.
3. Uses Symbols
   • Common symbols used include:
     • ∈ (belongs to),
     • ℝ (real numbers),
     • ℕ (natural numbers),
     • ℤ (integers),
     • : or | (such that).
4. Can Express Infinite Sets
   • Very useful when a set has an infinite number of elements.
   • Example: {x ∈ ℕ | x < 100} defines all natural numbers less than 100.
5. Logical Clarity
   • It clearly specifies what kind of elements are allowed in the set.

Applications of Set Builder Notation

1. Mathematical Problem Solving
   • Used to represent domains, solutions to equations, and conditions in math problems.
2. Computer Programming
   • Helps in defining sets, ranges, or filtering data in programming and databases.
3. Data Handling and Filtering
   • Useful in handling data sets where we apply conditions, like:
     {x | x is a student score > 75}
4. Graphing and Functions
   • Defines domain and range of functions.
   • Example: The domain of f(x) = 1/x can be written as {x ∈ ℝ | x ≠ 0}.
5. Set Theory and Logic
   • Plays a central role in formal definitions in set theory and symbolic logic.

Solved Examples of Set Builder Notation

Set builder notation is a representation that is commonly applied when the data set holds many or infinite numbers of elements. The common types of numbers involved with set builder questions are integers, real numbers, and natural numbers. Let us practice some solved examples to understand the same:

Example 1: Write the set builder form for the given set, P = { 12, 14, 16, 18, 20, 22, 24}.

Solution: The set builder notation for even numbers is as follows;

P = {a | a, is an even natural number that is greater than 11 and less than 25}

Example 2: What is the set builder notation for the below sets:

• The set of natural numbers from 1 to 10 (including 1 and 10).
• The set of numbers greater than or equal to 7
• The set of numbers less than or equal to 13
• The set of all even numbers
• The set of all odd numbers
• The set of all 12 months in a year

Solution:

1. The set of natural numbers from 1 to 10 (including both 1 and 10):
   Set: P = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
   Set builder notation:
   P = { p ∈ ℕ | 1 ≤ p ≤ 10 }
2. The set of numbers greater than or equal to 7:
   Set builder notation:
   Q = { q ∈ ℝ | q ≥ 7 }
3. The set of numbers less than or equal to 13:
   Set builder notation:
   R = { r ∈ ℝ | r ≤ 13 }
4. The set of all even numbers:
   Set builder notation:
   S = { s ∈ ℤ | s = 2n, n ∈ ℤ }
5. The set of all odd numbers:
   Set builder notation:
   A = { a ∈ ℤ | a = 2n + 1, n ∈ ℤ }
6. The set of all 12 months in a year:
   Set builder notation:
   B = { b | b is the name of a month of the year }

Learn about Universal Sets

Example 3: Decode the symbolic representations for the given examples. -13 ∉ N 3 ∈ W

Solution: -13 ∉ N

The statement -13 ∉ N says that -13 does not belong to the set of natural numbers.

3 ∈ W

The statement, 3 ∈ W denotes that 3 belongs to the set of whole numbers.

Example 4: Write the below examples in set builder notation. x ≤ 5 or x ≥ 8 {-8,-7,-6,-5,-4,-3, -2…}

Solution: x ≤ 5 or x ≥ 8

The above statement can be formulated as;

{x ∈ R | x ≤ 5 or x ≥ 8}

The statement {-8,-7,-6,-5,-4,-3, -2…} can be symbolised as;

A = { x | x > -9, x is an integer}

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

If you are checking Set builder notation article, also check related maths articles:
Roster notation	Cartesian product of sets
Difference between simple interest and compound interest	Irrational numbers
Fraction to percent	Natural numbers

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