Sets
O Level and IGCSE Mathematics Cheat Sheet – 1.2 Sets
Key Set Symbols and Meanings
| Symbol | Meaning | Example |
|---|---|---|
| n(A) | Number of elements in set A | If A = {1, 2, 3}, n(A) = 3 |
| ∈ | “is an element of” | 3 ∈ {1, 2, 3} |
| ∉ | “is not an element of” | 5 ∉ {1, 2, 3} |
| A′ | Complement of A (elements not in A) | If Universal set U = {1, 2, 3, 4, 5}, A = {1, 2}, then A′ = {3, 4, 5} |
| ∅ | Empty set (no elements) | ∅ = {} |
| Universal set (U) | All elements considered in a problem | For odd/even numbers ≤ 10, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} |
| A ⊆ B | A is a subset of B | {1, 2} ⊆ {1, 2, 3} |
| A ⊈ B | A is not a subset of B | {1, 4} ⊈ {1, 2, 3} |
| A ∪ B | Union: elements in A or B (or both) | {1, 2} ∪ {2, 3} = {1, 2, 3} |
| A ∩ B | Intersection: elements in both A and B | {1, 2} ∩ {2, 3} = {2} |
Describing Sets – Common Forms
- List form: D = {a, b, c, …}
- Rule form: A = {x : x is a natural number}
- Condition form: C = {x : a ≤ x ≤ b}
- Pair form: B = {(x, y) : y = mx + c}
Venn Diagram Basics
- Can be used for 2 sets or 3 sets.
- Circles represent sets, rectangle represents the universal set.
- Shading rules:
- A′ → Shade everything outside circle A.
- A ∩ B → Shade only the overlapping part of A and B.
- A ∪ B → Shade everything in either A or B.
- A′ ∩ B → Shade B excluding the overlap with A.
Quick Example – 2 Sets
U = {1, 2, 3, 4, 5, 6}
A = {1, 2, 3}, B = {3, 4, 5}
- A ∪ B = {1, 2, 3, 4, 5}
- A ∩ B = {3}
- A′ = {4, 5, 6}
- n(A ∩ B) = 1
Quick Example – 3 Sets
- In exam problems with 3 sets, fill in numbers starting from the centre (A ∩ B ∩ C) and work outwards.
- Always use given totals to fill missing values.
Exam Special Tips
- For subset questions, all elements of A must be in B to say A ⊆ B.
- Complement questions always require knowing the universal set.
- In number problems, use systematic filling in Venn diagrams to avoid missing elements.
- In set notation problems, order doesn’t matter for union/intersection: A ∪ B = B ∪ A, A ∩ B = B ∩ A.
Common Mistakes & Confusions
- Forgetting to exclude overlap when calculating numbers in separate sections.
- Writing elements outside of the universal set.
- Mixing up ∪ (union) and ∩ (intersection).
- Thinking ∅ means {0} — it does NOT. The empty set has no elements.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course