Set builder notation (or rule method) is a mathematical representation of a set by listing the elements or highlighting their common properties. Here, we ‘build’ the set by defining the logical properties of its elements.
A set of real numbers less than 8 is written in set builder notation as follows:
It is read as ‘the set of all x such that x is less than 8 and x belongs to real numbers.’ It can also be represented on interval notation when graphed on the number line:
There are three parts in a set when written in the set builder notation:
The variable is in lowercase.
‘Such that’ is expressed by the symbols ‘|’ (a vertical bar separator) or ‘:’ (colon).
Logical common properties of the elements are defined in the set.
These symbols are placed inside the curly brackets ‘{ }’ (set brackets).
Now, let us consider some more examples with different number types.
Set Builder Notation
Read as
Meaning
{x: x ≠ 0 and x ∈ ℤ} or {x | x ≠ 0 and x ∈ ℤ}
The set of all x such that x is not equal to 0 and x belongs to integers.
Any integers except 0
{x: x > 2 and x ∈ ℕ} or {x | x > 2 and x ∈ ℕ}
The set of all x such that x is greater than 2 and x belongs to natural numbers.
Any natural numbers greater than 2
{x: x ∈ ℚ and 1 < x < 2} or {x | x ∈ ℚ and 1 < x < 2}
The set of all x such that x belongs to rational numbers and x is between 1 and 2
Any rational number between 1 and 2
Why Use It
Set builder notation is used when there are many components, and the roster form makes it difficult to represent them.
Let us verify with an example by writing the natural numbers from 1 to 8 inclusively. We get {1, 2, 3, 4, 5, 6, 7, 8} by the roster notation.
Now, if we list all real numbers from 1 to 8 inclusively, the set becomes {1, 1.1, 1.01, 1.001, 1.0001, … } by the roster notation.
Writing such a set in the roster notation takes time and effort. On building the set in the set-builder notation, we get
{x | x is a real number, and 1 ≤ x ≤ 8} or {x | x is a rational or irrational number, and 1 ≤ x ≤ 8}
In addition to representing real numbers, rational numbers, and irrational numbers, the set builder notation is also used to represent intervals and equations. This notation makes the representation of sets easier, especially when they contain an unlimited number of components.
Defining Domain and Range
The set builder notation is quite helpful for defining the domain and range of a function. The domain is the set of all input values, and the range is the set of output values.
Now, let us find the domain and range of the rational function ${f\left( x\right) =\dfrac{1}{x+2}}$
When x = -2, the rational function f(x) is undefined, thus, the domain is all real integers except -2 and is written as {x | x ∈ ℝ and x ≠ -2}.
Now, by defining the range as
${y=\dfrac{1}{x+2}}$
⇒ ${x+2=\dfrac{1}{y}}$
⇒ ${x=\dfrac{1}{y}-2}$
Thus, the range of function in the set builder notation is written as {y | y ∈ ℝ and y ≠ 0}.
Here is the graph of the rational function ${f\left( x\right) =\dfrac{1}{x+2}}$:
Solved Examples
Use set builder notation to represent the following sets. a) A = {1, 8, 27, 64, 125, 216, 343, 512, 729, 1000} b) B = {0, 6, 12, 18, 24, 30, 36, …} c) C = {3, 4, 5}
Solution:
a) A = {x3 | x ∈ ℕ and x ≤ 10} b) B = {6x | x ∈ 𝕎} c) C = {x | 3 ≤ x ≤ 5 and x ∈ ℤ}
Use interval notation to represent the set in the set builder form that contains all the real values.
Solution:
The set containing all the real values is represented as {x | x ∈ ℝ} by the set builder notation. Now, in the interval notation, we get x ∈ (-∞, ∞) Thus, using the interval notation in the set builder form, we get the required set {x | x ∈ (-∞, ∞)}
