What is Set Builder Notation?
In Mathematics, set builder notation is a mathematical notation of describing a set by listing its elements or demonstrating its properties that its members must satisfy.
In set-builder notation, we write sets in the form of
{y | (properties of y)} OR {y : (properties of y)}
Where properties of y are replaced by the condition that completely describes the elements of the set. The symbol ‘|’ or ‘:’ is used to separate the elements and properties. The symbols ‘|’ or ‘:’ is read as “ such that” and the complete set is read as “ the set of all elements y” such that (properties of y). Here, we are using the variable ‘y’ to formulate the properties of the elements in the set.
Example:
X = {y: y is a letter in the word dictionary}
We read it as,
“X is the set of all y such that y is a letter in the word dictionary”.
What is Set in Mathematics?
In Mathematics, the set is an unordered group of elements represented by the sequence of elements (separated by commas) between curly braces {" and "}.
For example, {cat, cow, dog} is a set of domestic animals, {1, 3, 5, 7, 9} is a set of odd numbers, {a, b, c, d, e} is a set of alphabets.
Let Us Understand The Set Builder Notations
Set Builder Notations is the method to describe the set while describing the properties and not just listing its elements. When there is set formation in a set builder notation then it is called comprehension, set an intention, and set abstraction.
Set builder notation contains one or two variables and also defines which elements belong to the set and the elements which do not belong to the set. The rule and the variables are separated by slash and colon. This is often used for describing infinite sets.
Let Us Check Out The Symbols Used In Set Builder Notation
There are different symbols used for example for element symbol ∈ is denoted for element, the symbol ∉ is denoted to show that it is not an element, for the whole number it is W, symbol Z denotes integers, symbol N denotes all natural numbers and all the positive integers, symbol R denotes real numbers, symbol Q denotes rational numbers.
Set Builder Notation Symbols
The different symbols used to represent set builder notation are as follows:
∈: "Is an element of"
Example: x ∈ A means x is an element of the set A.∉: "Is not an element of"
Example: x ∉ A means x is not an element of the set A.∅ or {}: The empty set
A set with no elements. Example: ∅ = {}.W: Whole numbers
Usually W = { 0, 1, 2, 3, ... }.Z: Integers
Z = { ..., -2, -1, 0, 1, 2, ... }.N: Natural numbers
Usually N = { 1, 2, 3, ... }. Some definitions include 0, i.e., { 0, 1, 2, 3, ... }.R: Real numbers
Includes all numbers on the number line (rational and irrational).Q: Rational numbers
Numbers that can be expressed as fractions p/q, where p, q ∈ Z and q ≠ 0.C: Complex numbers
Numbers of the form a + bi, where a, b ∈ R and i² = -1.I or Irrational Numbers: Numbers that cannot be expressed as a fraction.
Example: √2, π, e.⊂: "Is a proper subset of"
Example: A ⊂ B means all elements of A are in B, but A ≠ B.⊆: "Is a subset of"
Example: A ⊆ B means all elements of A are in B, and A could equal B.∪: Union
The set containing all elements of A and B. Example: A ∪ B.∩: Intersection
The set containing elements common to both A and B. Example: A ∩ B.\ or −: Set difference
The set of elements in A but not in B. Example: A \ B or A − B.|: Such that
Used in set builder notation. Example: A = { x ∈ R | x > 0 } means A is the set of all x in R such that x > 0.∃: There exists
Example: ∃x ∈ A means there exists an x in A.∀: For all
Example: ∀x ∈ A means for all x in A.↔ or ⇔: If and only if
Example: P ↔ Q means P is true if and only if Q is true.⇒: Implies
Example: P ⇒ Q means if P is true, then Q is true.Z⁺ or N⁺: Positive integers
The set of positive whole numbers { 1, 2, 3, ... }.|x|: Absolute value
Represents the distance of x from 0 on the number line.{...}: Denotes a set explicitly
Example: A = { 1, 2, 3 }.∼: Similar to or equivalent to
Example: A ∼ B can mean A and B have a certain relation.
The set builder notation examples given below will help you to define set builder notation in the most appropriate way. The different set builder notation examples are as follows:
Set Builder Notation Examples
Representation of Sets Methods
There are two different methods to represent sets. These are:
Tabular Form or Roasted Method.
Set -Builder Form or Rule Method.
Tabular Form or Roasted Method
In the roaster method, the elements of the set are listed inside the braces {}, and each element is separated by commas. If the element appears more than once in the collection, it can be written only once.
Example,
The set X of the first five natural numbers is written as X = {1,2,3,4,5}.
The set A of the letter of the word MUMBAI is written as A = {M, U, B, A, I}.
Note: The elements of the set in the roasted method can be listed in any order. Hence, the set {A,B,C,D} can be written as {B, A, C,D}.
How do we Write A Set in Set Builder Method
If the elements of a set have a common property then they can be defined by describing the property. For example, the elements of the set A = {1,2,3,4,5,6} have a common property, which states that all the elements in the set A are natural numbers less than 7. No other natural numbers retain this property. Hence, we can write the set X as follows:
A = {x : x is a natural number less than 7} which can be read as “ A is the set of elements x such that x is natural numbers less than 7”.
The above set can also be written as A = {x : x N, x < 7}.
We can also write, set A = {the set of all the natural numbers less than 7}.
In this case, the description of the common property of the elements of a set is written inside the braces. This is the simple form of a set-builder form or rule method.
Set Builder Form Examples
Example1: Set of all even numbers: { x | x is an integer and x is even }
Meaning: The set contains all integers that are even.
Example 2: Set of all positive integers less than 10: { x | x is a positive integer and x < 10 }
Result: { 1, 2, 3, 4, 5, 6, 7, 8, 9 }
Example 3: Set of all real numbers greater than or equal to 0: { x ∈ R | x ≥ 0 }
Meaning: The set contains all real numbers 𝑥 that are greater than or equal to 0.
Example 4: Set of all prime numbers: { x ∈ N | x is prime }
Meaning: The set contains all natural numbers 𝑥 that are prime (e.g., { 2, 3, 5, 7, ... }).
Why do we Use Set Builder Notation?
If you are thinking why do we use such complicated notation to represent sets?
Or
What is the importance of using such complicated notation?
Now, you can find the answer to this question.
If you are asked to list a set of integers between 1 and 6, inclusive, then you can simply use a roaster form to write {1, 2, 3, 4, 5, 6}.
But the problem may raise if you will be asked to list the real numbers in the same interval in roaster from.
Using the set-builder notation would be convenient to use in this situation.
Starting with all the real numbers, we can limit them to the interval between 1 and 6 inclusive. Hence, it will be represented as:
{x : x ≥ 1 and x ≤ 6}
Set builder notation is also convenient to represent other algebraic sets. For example,
{y : y = y²}
Set-builder notation is widely used to represent infinite numbers of elements of a set.
Numbers such as real numbers, integers, natural numbers can be easily represented using the set-builder notation. Also, the set with an interval or equation can be best described by this method.
Set Builder Notation for Domain and Range
Set builder notation is a powerful way to describe the domain and range of a function by specifying the values that the input (domain) or output (range) can take.
1. Domain Examples
The domain of a function consists of all possible input values (usually x) for which the function is defined.
Example : Linear Function f(x)=2x+3
Domain: All real numbers because the function is defined for all x.
Set Builder Notation:{ x ∈ R | x is a real number }
2. Range Examples
The range of a function consists of all possible output values (usually f(x) or y) that the function can produce.
Example : Linear Function f(x)=2x+3
Range: All real numbers because the output can take any real value.
Set Builder Notation: { y ∈ R | y is a real number }
Set Builder Form Examples with Answers
1. Write the given set in the set-builder notation.
A = {1, 3, 5, 7, 9, 11, 13}
Solution: The given set A= {1, 3, 5, 7, 9, 11, 13} in the set-builder form can be written as:
{x : x is an odd natural numbers less than 14}.
2. How to write x ≤ 3 or x ≥ 4 in set-builder notation?
Solution: We can write x ≤ 3 or x ≥ 4 in set builder notation as:
{x ∈ R | x ≤ 3 or x ≥ 4}
FAQs on Set Builder Notation
1. What exactly is set-builder notation and why is it used in Maths?
Set-builder notation is a mathematical shorthand for describing a set by stating the properties that its elements must satisfy, rather than listing them all out. It's especially useful for sets with an infinite number of elements or for sets where the elements follow a specific rule, making it more efficient than the roster method.
2. What are the main parts of writing a set in set-builder form?
Set-builder notation has two main parts separated by a vertical bar (|) or a colon (:), which means 'such that'.
- Variable: The first part is a variable (like x, y, or n) that represents any element in the set.
- Rule: The second part is a rule or condition that the variable must satisfy to be a member of the set. For example, in {x | x is a prime number}, 'x' is the variable and 'is a prime number' is the rule.
3. What common symbols are needed to understand set-builder notation?
To read and write in set-builder notation, you should know these symbols:
- { }: Curly braces, used to enclose the set.
- | or : A vertical bar or colon, meaning 'such that'.
- ∈: 'is an element of' (e.g., x ∈ N means x is an element of the set of Natural numbers).
- N, W, Z, R: Symbols for Natural numbers, Whole numbers, Integers, and Real numbers, respectively.
4. How would you write the set A = {2, 4, 6, 8, 10} in set-builder form?
You can describe this set by its property, which is that all elements are even natural numbers up to 10. In set-builder notation, it can be written as:
A = {x | x is an even natural number and x ≤ 10}
Alternatively, you could write it as:
A = {x | x = 2n, n ∈ N and 1 ≤ n ≤ 5}
5. What is the main difference between the Roster Method and the Set-Builder Method?
The main difference is how they describe a set. The Roster Method simply lists all the individual elements of the set within curly braces, like {1, 2, 3}. In contrast, the Set-Builder Method describes the elements by stating a common property they all share, like {x | x is a natural number less than 4}.
6. When is it better to use set-builder notation instead of listing the elements?
Set-builder notation is much better when a set is very large or infinite. For example, it is impossible to list all positive real numbers in roster form. However, using set-builder notation, you can describe it easily as {x | x ∈ R, x > 0}. It's also better when the elements share a complex mathematical property.
7. Does the order of elements matter in a set? For example, is {1, 2, 3} the same as {3, 1, 2}?
No, the order of elements does not matter in a set. A set is defined only by the elements it contains, not their arrangement. Therefore, the set {1, 2, 3} is exactly the same as the set {3, 1, 2} because they both contain the same three elements.
8. How is set-builder notation used to describe the domain of a function?
Set-builder notation is very useful for defining the domain of a function, which is the set of all possible input values. For example, for the function f(x) = √x, the input 'x' cannot be negative. We can express its domain using set-builder notation as {x | x ∈ R, x ≥ 0}, which means 'the set of all real numbers x such that x is greater than or equal to 0'.
9. How does set-builder notation compare to interval notation?
Both are used to describe sets of numbers, but they look different. Interval notation uses parentheses ( ) and square brackets [ ] to show a range, like [2, 8], which means all numbers between 2 and 8, including 2 and 8. The equivalent in set-builder notation would be {x | x ∈ R, 2 ≤ x ≤ 8}. Set-builder notation is more versatile as it can describe sets that are not simple intervals, such as the set of all integers.
