Introduction to set theory/Lecture 1

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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.