Sets (Copy)
1. Understanding Sets
- Set: A collection of distinct objects, numbers, or symbols.
- Elements are listed within curly brackets
{ }. - Example: A = {2, 4, 6, 8}
- Elements are listed within curly brackets
- Element: An item in a set.
- Notation:
- If 3 is in set A → 3 ∈ A
- If 5 is not in set A → 5 ∉ A
- Notation:
- Example Set Definitions:
- A = {x : x is a natural number} → A = {1, 2, 3, 4, …}
- B = {a, b, c} → A set containing letters.
- C = {x : a ≤ x ≤ b} → A set of numbers between a and b inclusive.
2. Important Set Notation
| Symbol | Meaning | Example |
|---|---|---|
| n(A) | Number of elements in set A | If A = {2,4,6}, n(A) = 3 |
| A′ | Complement of A (everything not in A) | If ξ = {1,2,3,4}, A = {1,2} → A′ = {3,4} |
| ξ | Universal Set (all possible elements) | ξ = {1,2,3,4,5,6} |
| A ∪ B | Union: Elements in A or B | A = {1,2}, B = {2,3} → A ∪ B = {1,2,3} |
| A ∩ B | Intersection: Elements in both A and B | A = {1,2}, B = {2,3} → A ∩ B = {2} |
| ∅ | Empty Set | A = {} |
| ⊂ | Subset | A ⊂ B means all elements of A are in B |
3. Venn Diagrams (2 Sets Only)
- Universal Set (ξ): Represented by a rectangle containing all elements.
- Sets A and B: Represented by overlapping circles within the rectangle.
- Regions:
- Left circle = Set A
- Right circle = Set B
- Overlap = A ∩ B
- Outside both circles = Complement of A ∪ B, i.e., (A ∪ B)′
4. Key Concepts
- Union (A ∪ B):
All elements in A, in B, or in both.
Example:
A = {1, 3, 5}, B = {3, 4, 6}
A ∪ B = {1, 3, 4, 5, 6} - Intersection (A ∩ B):
Elements common to both sets.
Example:
A ∩ B = {3} - Complement (A′):
Elements in the Universal Set not in A.
Example:
If ξ = {1,2,3,4,5}, A = {2,4}
A′ = {1,3,5} - Number of Elements Formula:
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
5. Example Problems
Example 1:
If ξ = {1,2,3,4,5,6,7,8}
A = {2,4,6,8}
B = {1,2,3,4}
- A ∩ B = {2,4}
- A ∪ B = {1,2,3,4,6,8}
- A′ = {1,3,5,7}
Example 2:
n(A) = 10, n(B) = 8, n(A ∩ B) = 3
Find n(A ∪ B):
n(A ∪ B) = 10 + 8 – 3 = 15
6. Quick Reference Table
| Operation | Meaning | Example |
|---|---|---|
| A ∪ B | All elements in A or B | {1,2} ∪ {2,3} = {1,2,3} |
| A ∩ B | Elements common to both A and B | {1,2} ∩ {2,3} = {2} |
| A′ | Elements not in A | If ξ = {1,2,3}, A = {1} → A′ = {2,3} |
| n(A) | Number of elements in A | If A = {2,3,5}, n(A) = 3 |
| ∅ | Empty Set | No elements |
7. Tips for Exams
- Always start by filling the intersection in Venn diagrams.
- Use n(A ∪ B) formula to find missing values.
- Remember complements always refer to the Universal Set.
- For worded problems, define:
- Total = n(ξ)
- Place known values in correct Venn diagram regions.
- The complement of A ∪ B is the area outside both circles.