A set is a collection of well-defined objects that share common characteristics. In other words, a set is a group of objects with the same properties that helps to organise, interpret, and compare objects easily (such as a set of numbers, letters, or even real-life objects). The concept of sets is used everywhere, from elementary school math to computer science and real life. Therefore, it’s crucial to learn about types of sets, their representation, and formulas. In this article, we cover sets in detail, along with examples, types, and properties.
Table of Contents
A set is a well-defined group or collection of objects. These objects are called the elements or members of the set. The elements can be anything: numbers, letters, people, or even shapes.
Definition of Set:
A set is a collection of distinct elements, written within curly braces {} and separated by commas.
Example:
Set A = {2, 4, 6, 8}
This means set A contains 4 elements: 2, 4, 6, and 8.
In mathematics, a set is defined as a collection of different objects grouped and represented using curly braces {}. The members of a set are called its elements. For example, the set of natural numbers is represented as:
N = {1, 2, 3, 4,...}, where 1, 2, 3 are the elements of the set.
Z = {...-3, -2, -1, 0, 1, 2, 3,...}, where Z represents the set of integers.
The sets are represented in curly braces. The elements of a set are usually represented in either statement form, roaster form or set-builder form. Here are major ways and represention of sets in maths:
The standard or descriptive form is the way of defining a set without notations or listing elements. The set is defined with a description enclosed in curly braces.
Example:
O = {odd numbers less than 10} is a set of odd numbers less than 10 defined in standard form.
Roaster form or roaster notation is a way of listing all elements of a set inside curly braces, separated with a comma.
For Example:
A = {1, 3, 5}, Listing all elements in curly brackets.
C = {c, o, f, f, e, e}, C is the set of all letters in the word coffee.
In set-builder form, a notation or a rule, or a condition is given to define the elements.
Example:
A = {x | x is an odd number less than 6}, describing elements using a rule.
B = {x | x is a natural number less than 5}
This means: B = {1, 2, 3, 4}
Venn diagrams are helpful to visualize sets and how they relate. It is a graphical method used to show relationships between sets.
A single circle shows one set.
Overlapping circles show common elements.
Non-overlapping circles show disjoint sets.
Example:
Let A = {1, 2, 3}, B = {3, 4, 5}
Common element is 3 → It appears in the overlapping region.
Here are some common symbols used while working with sets:
Symbol |
Meaning |
Example |
∈ |
is an element of |
3 ∈ {1, 2, 3, 4} |
∉ |
is not an element of |
5 ∉ {1, 2, 3, 4} |
⊂ |
is a subset of |
{2, 4} ⊂ {2, 4, 6} |
⊄ |
is not a subset of |
{2, 5} ⊄ {1, 2, 3} |
∅ |
empty set or null set |
∅ = {} |
U |
universal set (all elements) |
U = {0 to 10} |
To truly define sets, we must understand their types:
Finite Set: Has a limited number of elements.
Example: {a, e, i, o, u}
Infinite Set: Goes on forever.
Example: {1, 2, 3, 4, ...}
A set with no elements.
Example: A = {} or A = ∅
A set with only one element.
Example: A = {0}
Two sets with the same elements.
Example: A = {1, 2}, B = {2, 1} → A = B
A set formed from elements of another set is known as a Subset.
If B = {2, 4} and A = {2, 4, 6}, then B ⊂ A
A set that contains all elements under consideration is called a universal set.
Example: If discussing numbers 1 to 10, then U = {1, 2, ..., 10}
Here are the main operations you can perform with sets:
Union of two sets A and B is a set of all elements combined from both sets.
Example:
A = {1, 2}, B = {2, 3}
A ∪ B = {1, 2, 3}
Intersection of sets is a set that contains only the common elements present in both sets.
Example:
A = {1, 2, 3}, B = {2, 3, 4}
A ∩ B = {2, 3}
Difference of set A and B is a set with elements that are present in A but not B.
Example:
A = {1, 2, 3}, B = {2, 3, 4}
A − B = {1}
Complement of a set is set that contains the elements that are present not in the set but in the universal set.
If U = {1 to 5}, A = {2, 3},
Then A’ = {1, 4, 5}
Let’s understand how sets are used in practical problems:
Problem:
In a class, 15 students like Maths, 10 like Science, and 5 like both.
How many students like only one subject?
Solution:
Only Maths = 15 − 5 = 10
Only Science = 10 − 5 = 5
Total = 10 + 5 = 15 students
Write the set of vowels in English.
List even numbers from 1 to 10 in set-builder form.
Find A ∩ B for A = {1, 3, 5}, B = {3, 4, 5}
If U = {1 to 10}, A = {2, 4, 6}, find A’
Create a Venn diagram showing A = {1, 2}, B = {2, 3}
While sets are just one concept in mathematics to learn, it is impossible to succeed in the subject without understanding them. Learning how to define sets, their types, union, intersection and formulas is essential not only for math but other subjects too. Venn diagrams used to represent the union or intersection of sets, prepares us in the skill to learn higher mathematics and work on different tasks independently.
Ans: In mathematics, a set is a collection of distinct and well-defined objects or elements. These objects can be numbers, letters, or even items, and they are usually written inside curly brackets like {1, 2, 3}.
Ans: The universal set includes all elements under consideration for a particular discussion. If we are talking about numbers from 1 to 10, then the universal set is written as:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Ans: Here are 12 types of sets in mathematics:
Empty Set
Finite Set
Infinite Set
Equal Set
Unequal Set
Subset
Proper Subset
Improper Subset
Universal Set
Power Set
Singleton Set
Disjoint Set
Ans: The Principle of Inclusion-Exclusion for two sets or Union-Intersection formula is used to find the total number of elements in two overlapping sets A and B.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Ans: "5 sets" usually means a group of 5 different sets. For example:
Set A = {1, 2}
Set B = {3, 4}
Set C = {5}
Set D = {6, 7, 8}
Set E = {9, 10}
These are five different sets, each with its own elements
Want to learn more about how sets connect with geometry, algebra, and real-world data?
Explore more math concepts at Orchids The International School .
CBSE Schools In Popular Cities