Sets are one of the fundamental concepts in mathematics. A set is simply a collection of objects, called elements or members. Sets help us organize and classify objects based on their properties. In this guide, we’ll explore the basic concepts of sets, different types of sets, set notation, and how to represent sets using Venn diagrams.
1. What is a Set? #
A set is a well-defined collection of distinct objects. These objects can be anything: numbers, people, letters, colors, or even other sets!
Key Properties of Sets #
- Order doesn’t matter – The elements in a set have no particular order
- No repetition – Each element appears only once in a set
- Well-defined – It must be clear whether an object belongs to the set or not
Examples: #
- A set of even numbers less than 10: {2, 4, 6, 8}
- A set of vowels in the English alphabet: {a, e, i, o, u}
- A set of colors in a rainbow: {red, orange, yellow, green, blue, indigo, violet}
2. Set Notation #
Set notation gives us a standard way to write and describe sets, making them easier to work with in mathematical expressions.
Ways to Represent Sets #
1. Roster or Enumeration Method #
Listing all elements inside curly braces { }.
Example: A = {1, 2, 3, 4, 5}
2. Set-Builder Notation #
Describing the elements using a rule or property.
Example: A = {x | x is a natural number less than 6}
This is read as “A is the set of all x such that x is a natural number less than 6”
3. Using Capital Letters #
Sets are usually denoted by capital letters (A, B, C, …)
Elements are usually denoted by lowercase letters (a, b, c, …)
Key Symbols in Set Notation #
| Symbol | Meaning | Example |
|---|---|---|
| ∈ | Belongs to / is an element of | 3 ∈ {1, 2, 3, 4} |
| ∉ | Does not belong to | 5 ∉ {1, 2, 3, 4} |
| { } | A set (enclosed in curly braces) | A = {1, 2, 3} |
| | | “such that” (used in set-builder notation) | {x | x > 5} |
| ⊂ | Subset (every element of A is also in B) | {1, 2} ⊂ {1, 2, 3, 4} |
| ⊆ | Subset or equal to | {1, 2, 3} ⊆ {1, 2, 3} |
| ∪ | Union | A ∪ B |
| ∩ | Intersection | A ∩ B |
| A’ | Complement of set A | A’ or Ac |
| ∅ | Empty set or null set | ∅ or { } |
| |A| | Cardinality (number of elements in set A) | |{1, 2, 3}| = 3 |
3. Different Types of Sets #
Sets can be classified into different types based on their elements and properties. Understanding these types helps in solving various mathematical problems.
Universal Set (U or ξ) #
The set of all elements under consideration in a particular problem or context.
Example: If we’re discussing colors, the universal set might be all colors that exist.
Empty Set (∅ or { }) #
A set that contains no elements.
Example: The set of even numbers that are divisible by 2 and end in 3.
Infinite Set #
A set with an uncountable or infinite number of elements.
Example: The set of all natural numbers {1, 2, 3, …}
Subset #
A set A is a subset of set B if every element of A is also an element of B.
We write this as A ⊂ B
Example: If A = {1, 2} and B = {1, 2, 3, 4}, then A ⊂ B
Proper Subset #
A is a proper subset of B if A is a subset of B, but A ≠ B.
Example: {1, 2} is a proper subset of {1, 2, 3}
Equal Sets #
Two sets A and B are equal if they contain exactly the same elements.
Example: {1, 2, 3} = {3, 1, 2} (order doesn’t matter)
Disjoint Sets #
Sets that have no elements in common.
Example: {1, 3, 5} and {2, 4, 6} are disjoint sets
Common Number Sets #
- ℕ – Natural numbers: {1, 2, 3, …}
- ℤ – Integers: {…, -2, -1, 0, 1, 2, …}
- ℚ – Rational numbers (fractions)
- ℝ – Real numbers (all numbers on the number line)
Set Operations #
Union (∪) #
The union of sets A and B (written A ∪ B) is a set containing all elements that are in A or B or both.
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}
Intersection (∩) #
The intersection of sets A and B (written A ∩ B) is a set containing only elements that are in both A and B.
Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}
Complement (A’) #
The complement of set A (written A’ or Ac) is a set containing all elements in the universal set that are not in A.
Example: If U = {1, 2, 3, 4, 5} and A = {1, 3, 5}, then A’ = {2, 4}
Difference (A – B) #
The difference of sets A and B (written A – B) is a set containing elements that are in A but not in B.
Example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A – B = {1, 2}
Symmetric Difference (A Δ B) #
The symmetric difference of sets A and B is a set containing elements that are in either A or B, but not in both.
Example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then A Δ B = {1, 2, 5, 6}
4. Venn Diagrams #
Venn diagrams use circles to visually represent sets and their relationships. They make it easier to understand set operations and solve problems involving sets.
Basic Venn Diagram Components #
- Rectangle – Represents the universal set (U)
- Circles – Represent individual sets
- Overlapping regions – Represent elements common to multiple sets
Using Venn Diagrams to Solve Problems #
Venn diagrams are particularly useful for solving problems involving set operations and counting elements.
Example: #
In a class of 35 students:
- 20 study Mathematics
- 15 study Physics
- 10 study both Mathematics and Physics
How many students study neither Mathematics nor Physics?
Solution:
Let’s denote:
- M = set of students studying Mathematics
- P = set of students studying Physics
Given:
- |M| = 20
- |P| = 15
- |M ∩ P| = 10
- Total students = 35
Number of students studying Mathematics only = |M| – |M ∩ P| = 20 – 10 = 10
Number of students studying Physics only = |P| – |M ∩ P| = 15 – 10 = 5
Number of students studying either Mathematics or Physics = 10 + 10 + 5 = 25
Therefore, number of students studying neither = 35 – 25 = 10
Set Formulas Using Venn Diagrams #
- n(A ∪ B) = n(A) + n(B) – n(A ∩ B) – To find the total number of elements in the union of two sets
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(A ∩ C) – n(B ∩ C) + n(A ∩ B ∩ C) – For three sets
- n(U) = n(A) + n(A’) – The universal set equals a set plus its complement
5. Set Problems and Techniques #
The following techniques and approaches will help you solve problems involving sets.
Common Problem Types #
- Finding elements in set operations (union, intersection, etc.)
- Counting problems using Venn diagrams
- Proving set equalities using algebraic methods
- Verifying subset relationships
General Approach to Solving Set Problems #
- Identify the sets involved and the information given
- Draw a Venn diagram if appropriate
- Fill in known information (from outside to inside)
- Use set formulas to find missing values
- Check your answer makes logical sense
Example Problem: #
In a survey of 100 people, it was found that:
- 65 people like tea (T)
- 58 people like coffee (C)
- 37 people like both tea and coffee
Find the number of people who like:
- Either tea or coffee
- Tea but not coffee
- Neither tea nor coffee
Solution:
a) Number of people who like either tea or coffee:
n(T ∪ C) = n(T) + n(C) – n(T ∩ C) = 65 + 58 – 37 = 86
b) Number of people who like tea but not coffee:
n(T) – n(T ∩ C) = 65 – 37 = 28
c) Number of people who like neither tea nor coffee:
n(U) – n(T ∪ C) = 100 – 86 = 14
