Introduction to set theory/Lecture 1
IntroductionEdit
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.
NotationsEdit
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 TableEdit
Truth tables are used to analyze formulae of propositional logic.
ExampleEdit
Truth table for
| T | T | T | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |
TautologyEdit
DefinitionEdit
A formula of propositional logic is a tautology if only T's occur in the column of the truth table.
ExamplesEdit
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 EquivalenceEdit
DefinitionEdit
The proposition formulas and are tautologically equivalent if is a tautology.
ExamplesEdit
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 ResourcesEdit
The materials in this course overlap with Introductory Discrete Mathematics for Computer Science, particularly Lesson 1.