Set Builder Notation

Set-builder notation is a representation used to write and represent the elements of sets, often for sets with an infinite number of elements. It is used with common types of numbers, such as integers, real numbers, and natural numbers. This set-builder notation is also used to express sets with an interval or an equation.

Let us learn more about the symbols used in set builder notation, the domain and range, the uses of set-builder notation, with the help of examples, FAQs.

Set-builder notation is defined as a representation or a notation that can be used to describe a set that is defined by a logical formula that simplifies to be true for every element of the set. The set builder notation includes one or more than one variables. It also defines a rule about the elements which belong to the set and the elements that do not belong to the set. Let us read about different methods of writing sets.

Methods of Writing Set

There are two methods that can be used to represent a set. The roster form or listing the individual elements of the sets, and the set builder form of representing the elements with a statement or an equation. The two methods are as follows.

Roster Form or Listing Method

In this method, we list down all the elements of a set, and they are represented inside curly brackets. Each of the elements is written only once and is separated by commas. For example, the set of letters in the word, "California" is written as A = {c, a, l, i, f, o, r, n}.

Set Builder Form or Rule Method 

Set builder form uses a statement or an expression to represent all the elements of a set. In this method, we do not list the elements; instead, we will write the representative element using a variable followed by a vertical line or colon and write the general property of the same representative element.

Here are some set builder notation form examples.

Example:

A = {x | x ∈ N, 5 < x < 10} and is read as "set A is the set of all ‘x’ such that ‘x’ is a natural number between 5 and 10."

The symbol ∈ means “is an element of” and denotes membership of an element in a set.

Example:

B = { x | x is an odd number between 11 and 20} which means set B contains all the odd numbers between 11 and 20. By using the roster method, set B can be written as B = {11, 13, 15, 17, 19}. Q is the set of rational numbers that can be written in set builder form as Q={p/q | p, q ∈ I, q≠0}. The above is read as ‘Q’ is the set of all numbers in the form q/p such that p and q are integers where q is not equal to zero.’

The set builder notation uses various symbols to represent the elements of the set. A few of the symbols are listed as follows. 

• ∈ means "is an element of".
• ∉ means "is not an element of".
• N represents natural numbers or all positive integers.
• W represents whole numbers.
• Z indicates integers.
• Q represents rational numbers or any number that can be expressed as a fraction.
• R represents real numbers or any number that isn't imaginary.

Set builder notation is used when there are numerous elements and we are not able to easily represent the elements of the set by using the roster form. Let us understand this with the help of an example. If you have to list a set of integers between 1 and 8 inclusive, one can simply use roster notation to write {1, 2, 3, 4, 5, 6, 7, 8}. But the problem arises when we have to list the real numbers in the same interval. Using roster notation would not be practical. {1, 1.1, 1.01, 1.001, 1.0001, ... ??? }. But using the set-builder notation would be better in this scenario. Starting with all real numbers, we can limit them to the interval between 1 and 8 inclusive.{x|x≥1{x|x≥1 and x≤8}. It is quite convenient to use set builder notation to express other algebraic sets, such as: {x|x=x²}.

Set-builder notation comes in handy to write sets, especially for sets with an infinite number of elements. Numbers such as integers, real numbers, and natural numbers can be expressed using set-builder notation. A set with an interval or an equation can also be expressed using this method.

Set builder notation is very useful for defining the domain and range of a function. In its simplest form, the domain is the set of all the values that go into a function. For Example: For a function, f(x) = 2/(x-1) the domain would be all real numbers, except +1. This is because the function f(x) would be undefined when x = 1. Thus, the domain for the above function can be expressed as {x∈R|x≠1}.

Set Builder Notation and Interval Notation

Set builder notation is represented as Interval notation, and it is a way to define a set of numbers between a lower limit and an upper limit using end-point values. The upper and lower limits may or may not be included in the set. The end-point values are written between brackets or parentheses. A square bracket denotes inclusion in the set, while a parenthesis denotes exclusion from the set. For example, (4,12]. This interval notation denotes that this set includes all real numbers between 4 and 12. 12 is included in the set while 4 is not a part of the set. Suppose we want to express the set of real numbers {x |-2 < x < 5} using an interval. This can be expressed as interval notation (-2, 5).

The set of real numbers can be expressed as (-∞, ∞).

☛Related Articles

Check out a few more articles closely connected to the set builder Notation for a better understanding of the topic.

• Operations on Sets
• Venn Diagrams
• Subset
• Roster Notation
• Universal Set
• Intersection of Sets

What is Set-Builder Notation in Math?

Set-builder notation is a mathematical notation for describing a set by representing its elements or explaining the properties that its members must satisfy. For example, C = {2,4,5} denotes a set of three numbers: 2, 4, and 5, and D ={(2,4),(−1,5)} denotes a set of two pairs of numbers. Another option is to use set-builder notation: F = {n³: n is an integer with 1≤n≤100} is the set of cubes of the first 100 positive integers.

What is Set Builder Notation Form Example?

A set-builder notation describes the elements of a set instead of listing the elements. For example, the set { 5, 6, 7, 8, 9} list the elements. We read the set {x is a counting number between 4 and 10} as the set of all x such that x is a number greater than 4 and less than 10.

How do you Express Intervals in Set Builder Notation?

For the given set A = {..., -3, -2, -1, 0, 1, 2, 3, 4}. A = {x ∈ Z | x ≤ 4 }.

How do you Write Inequalities in Set Builder Notation?

The inequalities in sets builder notation is written using >, <, >, <, symbols. { x | x ∈ R, x ≥ 2 and x ≤ 6 }. This indicates that the set includes all the real numbers, between 2 and 6 inclusive.

How do you Write Domain in Set Builder Notation Form?

We can write the domain of f(x) in set builder notation as, {x | x ≥ 0}. If the domain of a function is all real numbers we can state the domain as, 'all real numbers,'. Also, we can use the symbol to represent all real numbers.

What is Interval Notation and Set Builder Notation Form?

In the Interval notation, the end-point values are written between brackets or parentheses. A square bracket denotes inclusion in the set, while a parenthesis denotes exclusion from the set. For example, (8,12]. This interval notation denotes that this set includes all real numbers between 8 and 12. Whereas set-builder notation is a mathematical notation for describing a set by representing its elements or explaining the properties that its members must satisfy. For example, For the given set A = {..., -3, -2, -1, 0, 1, 2, 3, 4}. A = {x ∈ Z | x ≤ 4 }.