Sets
Complete Mathematics Notes
Learning Goals
What you will learn in this chapter
After studying this chapter, you should be able to:
- Understand set notation and symbols.
- Identify different types of sets.
- Represent sets in different forms.
- Perform operations on sets.
- Understand Venn diagrams and cardinality.
- Solve set-related word problems for SEE preparation.
Meaning and Notation
Understanding the basic concept of a set
A set is a well-defined collection of distinct objects. These objects are called elements or members of the set.
A = {2, 4, 6, 8}
Important Set Symbols
Belongs to
2 ∈ A means 2 is an element of A.
Does not belong to
3 ∉ A means 3 is not an element of A.
Empty Set
A set that contains no elements.
Subset
Every element of one set belongs to another set.
- Sets are usually represented by capital letters.
- Elements are enclosed within curly braces { }.
- The order of elements does not matter.
- Repeated elements are counted only once.
Examples
Example 1: The order of elements does not matter.
Example 2: Repeated elements count once.
Ways to Describe Sets
Three common methods of representing sets
Roster or Tabular Form
List all the elements inside curly braces.
Set-Builder Form
Describe the property satisfied by the elements.
The colon means “such that”.
Descriptive Form
Describe the set using words.
Common Types of Sets
Definitions and examples for quick revision
| Type of Set | Meaning | Example |
|---|---|---|
| Empty / Null Set | A set containing no elements. | ∅ or { } |
| Singleton Set | A set containing exactly one element. | {0} |
| Finite Set | A set having a limited number of elements. | {a, e, i, o, u} |
| Infinite Set | A set having unlimited elements. | Natural numbers |
| Equal Sets | Sets containing exactly the same elements. | {1, 2} and {2, 1} |
| Equivalent Sets | Sets containing the same number of elements. | {a, b, c} and {4, 5, 6} |
| Subset | Every element of A is also in B. | {1, 2} ⊆ {1, 2, 3} |
| Universal Set | All objects under discussion. | U = {1, 2, 3, 4, 5} |
Operations on Sets
Four essential operations to understand
Union
A ∪ B
The union contains all elements in A, B, or both.
B = {3, 4}
A ∪ B = {1, 2, 3, 4}
Intersection
A ∩ B
The intersection contains elements common to both sets.
B = {3, 4}
A ∩ B = {3}
Sets with no common elements are called disjoint sets.
Difference
A − B
The difference contains elements in A but not in B.
B = {3, 4}
A − B = {1, 2}
Complement
A′
The complement contains elements of U that are not in A.
A = {1, 2, 3}
A′ = {4, 5}
Cardinality and Formulas
Important formulas for examination revision
Cardinality means the number of distinct elements in a finite set.
Union Formula
Subtract the intersection because it is counted twice.
Disjoint Sets
Applies when A ∩ B = ∅.
Complement Formula
Used to find the number of elements outside A.
Other Important Laws
Union with the empty set
Intersection with the empty set
Idempotent law of union
Idempotent law of intersection
Commutative law of union
Commutative law of intersection
Worked Example
A typical SEE-style word problem
In a class of 40 students, 23 like football, 18 like basketball, and 7 like both games. Find:
- The number of students who like at least one game.
- The number of students who like neither game.
Solution
Write the given information.
n(F) = 23
n(B) = 18
n(F ∩ B) = 7
Find the number who like at least one game.
= 23 + 18 − 7
= 34 students
Find the number who like neither game.
= 40 − 34
= 6 students
Adding 23 and 18 counts the 7 students who like both games twice. Subtract 7 once to count everyone correctly.
Exam Tips
Remember these points when solving questions
Use the Correct Formula
For union questions, subtract the intersection once.
Draw Venn Diagrams
Fill the intersection first, then the remaining regions.
Read Carefully
Identify only A, both, at least one, or neither.
Check the Universal Set
The complement depends on the universal set U.
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