📚 Set Theory – Quick Study Notes (Quantitative Ability)
Set theory is the foundation of many quantitative problems.
It deals with collections of objects, called sets, and the relationships between them.
1. Simple Explanation
| Term | Meaning | Symbol | Example |
|---|---|---|---|
| Set | A collection of distinct objects (called elements). | (A, B, C, \dots) | (A = {1, 2, 3}) |
| Element | An item inside a set. | (x \in A) | (2 \in A) |
| Subset | Every element of set (B) is in set (A). | (B \subseteq A) | ({1,2} \subseteq {1,2,3}) |
| Empty set | A set with no elements. | (\emptyset) | ({x |
| Universal set | The set containing every element under consideration. | (U) | In a problem about natural numbers (1–10), (U = {1,2,\dots,10}) |
| Union | All elements that are in either set. | (A \cup B) | ({1,2,3}\cup{3,4} = {1,2,3,4}) |
| Intersection | Elements common to both sets. | (A \cap B) | ({1,2,3}\cap{3,4} = {3}) |
| Difference | Elements in (A) but not in (B). | (A – B) | ({1,2,3} - {2} = {1,3}) |
| Complement | Elements not in the set (relative to (U)). | (A^c) | If (U={1,2,3,4}), (A={1,3}), then (A^c={2,4}) |
| Cardinality | Number of elements in a set. | ( | A |
2. Key Points (What You Must Remember)
| # | Key Point | Why It Matters |
|---|---|---|
| 1 | Always define the universal set before doing operations. | Without (U), complements are ambiguous. |
| 2 | Union & Intersection are commutative & associative. | Helps rearrange expressions freely. |
| 3 | De Morgan’s Laws: ((A \cup B)^c = A^c \cap B^c) & ((A \cap B)^c = A^c \cup B^c). | Useful for simplifying complements. |
| 4 | Inclusion–Exclusion Principle: ( | A \cup B |
| 5 | Cardinality of a power set: ( | P(A) |
| 6 | Venn Diagrams: Visual tool for small sets (≤ 3). | Great for checking your logic. |
| 7 | Handling complements: Always refer back to (U). | Prevents mistakes in “not in” questions. |
3. Worked Example
Problem
In a class of 30 students,
- 18 students study Mathematics (M).
- 12 students study Physics (P).
- 8 students study both Mathematics and Physics (M ∩ P).
How many students study neither subject?
Step‑by‑Step Solution
| Step | Operation | Reasoning |
|---|---|---|
| 1 | Define (U =) all 30 students. | Universal set. |
| 2 | Find ( | M \cup P |
| 3 | Compute: ( | M \cup P |
| 4 | Find complement: ( | (M \cup P)^c |
| 5 | Compute: ( | (M \cup P)^c |
Answer: 8 students study neither Mathematics nor Physics.
Venn Diagram (Optional)
┌───────────────┐
│ M ∩ P (8) │
┌─────┴─────┬─────┴─────┐
│ M only │ P only │
│ (10) │ (4) │
└───────────┴──────────┘
- M only = 18 – 8 = 10
- P only = 12 – 8 = 4
- Neither = 30 – (10 + 4 + 8) = 8
4. Quick Revision Summary
- Set notation: (\in, \subseteq, \cup, \cap, -, ^c).
- Universal set (U) is the reference for complements.
- De Morgan’s Laws help switch between unions and intersections under complement.
- Inclusion–Exclusion is the go‑to formula for counting overlaps.
- Cardinality (|A|) = number of elements; (|P(A)| = 2^{|A|}).
- Venn Diagrams: Use for 2–3 sets to visualize.
- **Always