Complete Guide to Sets
1.1 Introduction to Sets
A set is a well-defined collection of distinct objects, called elements or members.
Examples:
- Set of vowels: {a, e, i, o, u}
- Set of natural numbers: N = {1, 2, 3, ...}
Sets are fundamental building blocks in mathematics used to define relations, functions, and other structures.
1.2 Sets and their Representations
Sets can be represented in three ways:
| Method | Description | Example |
|---|---|---|
| Roster/Tabular Form | List all elements between braces | {2, 4, 6, 8} |
| Set-Builder Form | Describe properties of elements | {x | x is an even number less than 10} |
| Descriptive Form | Describe the set in words | "Set of even numbers less than 10" |
1.3 The Empty Set
The empty set (or null set) is a set containing no elements, denoted by ∅ or {}.
Examples:
- Set of months with 32 days: ∅
- Solution to x² + 1 = 0 in real numbers: ∅
Note: {∅} is NOT an empty set - it's a set containing one element (the empty set).
1.4 Finite and Infinite Sets
| Type | Definition | Example |
|---|---|---|
| Finite Set | Has countable number of elements | {1, 2, 3, 4, 5} |
| Infinite Set | Has unlimited number of elements | {1, 2, 3, ...} |
Practice: Identify finite and infinite sets:
- Set of planets in our solar system (Finite)
- Set of points on a line (Infinite)
1.5 Equal Sets
Two sets A and B are equal (A = B) if they contain exactly the same elements, regardless of order or repetition.
Examples:
- {1, 2, 3} = {3, 2, 1} = {1, 1, 2, 3}
- {a, b} ≠ {a, b, c}
1.6 Subsets
A set A is a subset of B (A ⊆ B) if every element of A is also in B.
Venn Diagram Representation:
◯ A inside ◯ B
Examples:
- {1, 2} ⊆ {1, 2, 3}
- ∅ ⊆ A for any set A (empty set is subset of all sets)
Proper Subset (A ⊂ B): A is subset of B but A ≠ B.
1.7 Universal Set
The universal set (U) is the set containing all objects under consideration.
Examples:
- When discussing numbers, U might be all real numbers
- In a class survey, U might be all students in the class
Venn diagrams represent U as a rectangle enclosing all other sets.
1.8 Venn Diagrams
Venn diagrams use overlapping circles to visually represent relationships between sets.
Basic Venn Diagram Components:
Rectangle: Universal set (U)
Circles: Individual sets
Overlapping areas: Common elements
Example: For sets A = {1, 2, 3} and B = {3, 4, 5}
Venn diagram shows two overlapping circles with:
- Left circle: 1, 2
- Overlap: 3
- Right circle: 4, 5
1.9 Operations on Sets
| Operation | Notation | Definition | Example |
|---|---|---|---|
| Union | A ∪ B | Elements in A or B or both | {1,2} ∪ {2,3} = {1,2,3} |
| Intersection | A ∩ B | Elements common to both A and B | {1,2} ∩ {2,3} = {2} |
| Difference | A - B | Elements in A but not in B | {1,2} - {2,3} = {1} |
1.10 Complement of a Set
The complement of set A (A' or Ac) contains all elements in U that are not in A.
Venn Diagram: Entire rectangle except circle A
Example: If U = {1,2,3,4,5} and A = {1,2}, then:
A' = {3,4,5}
Key Property: A ∪ A' = U and A ∩ A' = ∅