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