📦 What are Sets?
A Set is a well-defined collection of distinct objects, considered as a single entity. Sets are fundamental in all areas of mathematics.
Objects in a set are called elements or members.
Notation: Sets are usually written with curly brackets:
{1, 2, 3}
🔤 Types of Sets
- Empty Set (∅): A set with no elements. Example: { } or ∅
- Finite Set: Set with a countable number of elements.
- Infinite Set: Set with uncountable elements. Example: {1, 2, 3, ...}
- Subset: A set A is a subset of B if every element of A is in B. Notation: A ⊆ B
- Proper Subset: A ⊂ B if A ⊆ B and A ≠ B
- Universal Set: The set that contains all elements under discussion. Usually denoted by U.
- Power Set: The set of all subsets of a set. If A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}
- Disjoint Sets: Two sets that have no elements in common.
⚙️ Set Operations
- Union (A ∪ B): Elements in A, B, or both
- Intersection (A ∩ B): Elements common to both A and B
- Difference (A - B): Elements in A but not in B
- Complement (A′): Elements not in A (within the universal set)
Example: Let A = {1, 2, 3}, B = {2, 3, 4}
- A ∪ B = {1, 2, 3, 4}
- A ∩ B = {2, 3}
- A - B = {1}
- B - A = {4}
📊 Venn Diagrams
Venn diagrams are visual representations of sets and their relationships using overlapping circles.
Basic Shapes:
- Circle A → Represents Set A
- Overlap of A and B → A ∩ B
- Entire shaded area of both circles → A ∪ B
- Area in A only → A - B
🧠 Introduction to Logic
Logic is the framework that determines truth in mathematical reasoning. It involves statements, operators, and rules for combining them.
Statements (Propositions)
A statement is a sentence that is either true or false — not both.
Examples:
- 2 + 2 = 4 → True
- The sky is green → False
🔗 Logical Operators
- Negation (¬p): "Not p"
- Conjunction (p ∧ q): "p and q" → True only if both are true
- Disjunction (p ∨ q): "p or q" → True if at least one is true
- Implication (p → q): "If p, then q"
- Biconditional (p ↔ q): "p if and only if q"
Example Table (Truth Table):
p | q | p ∧ q | p ∨ q | p → q | p ↔ q
-------------------------------------
T | T | T | T | T | T
T | F | F | T | F | F
F | T | F | T | T | F
F | F | F | F | T | T
🛠 Logical Equivalences
Used to simplify logical expressions. Two statements are equivalent if they have the same truth values in all cases.
- De Morgan’s Laws:
- ¬(p ∧ q) ≡ ¬p ∨ ¬q
- ¬(p ∨ q) ≡ ¬p ∧ ¬q
- Double Negation: ¬(¬p) ≡ p
- Contrapositive: p → q ≡ ¬q → ¬p
📘 Applications of Sets and Logic
- Database querying (SQL logic)
- Computer science: algorithms, programming, binary logic
- Mathematical proofs
- Venn diagrams in data science
- Set-based decision-making systems
🧪 Practice Problems
- List the power set of A = {1, 2}
- Let U = {1,2,3,4,5}, A = {1,2,3}, B = {3,4}. Find A ∪ B, A ∩ B, and A′.
- Draw a Venn diagram for sets A and B with A ∩ B ≠ ∅
- Use truth tables to determine if (p ∧ q) → p is always true
- Simplify ¬(p ∨ q) using De Morgan's law