Introduction to set theory/Lecture 1
Introduction edit
We will start the course by introducing Propositional Logic. Even though this is a set theory class and not a logic course, most of notations from the logic courses can be used in set theory. Furthermore, logic is important in various proofs we will encounter in this course.
Notations edit
Here are the notations and what they mean:
| Symbols | Meaning |
|---|---|
| and (conjunction) | |
| or (nonexclusive disjunction) | |
| not (negation) | |
| if then/implies | |
| if and only if |
Truth Table edit
Truth tables are used to analyze formulae of propositional logic.
Example edit
Truth table for
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |
Tautology edit
Definition edit
A formula of propositional logic is a tautology if only T's occur in the column of the truth table.
Examples edit
Truth table for
| T | F | F | T |
| F | T | T | T |
Truth table for
| T | T | F | F | F | F | T |
| T | F | T | F | F | F | T |
| F | T | F | F | T | T | T |
| F | F | T | F | T | T | T |
Truth table for
| T | T | F | T | T | T |
| T | F | F | F | F | T |
| F | T | T | T | T | T |
| F | F | T | T | T | T |
Tautological Equivalence edit
Definition edit
The proposition formulas and are tautologically equivalent if is a tautology.
Examples edit
Contraposition: is tautologically equivalent to .
| T | T | F | F | T | T | T |
| T | F | T | F | F | F | T |
| F | T | F | T | T | T | T |
| F | F | T | T | T | T | T |
de Morgan's Law I: is tautologically equivalent to .
| T | T | F | F | T | F | F | T |
| T | F | F | T | T | F | F | T |
| F | T | T | F | T | F | F | T |
| F | F | T | T | F | T | T | T |
de Morgan's Law II: is tautologically equivalent to . Truth table for Assignment #1
Related Resources edit
The materials in this course overlap with Introductory Discrete Mathematics for Computer Science, particularly Lesson 1.