Introduction to set theory/Lecture 1

From Wikiversity
Jump to: navigation, search


Introduction[edit]

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.

Notations[edit]

Here are the notations and what they mean:

Symbols Meaning
\land and (conjunction)
\lor or (nonexclusive disjunction)
\lnot not (negation)
\to if then/implies
\leftrightarrow if and only if

Truth Table[edit]

Truth tables are used to analyze formulae of propositional logic.

Example[edit]

Truth table for p \to (q \to p)

p \,\! q \,\! q \to p p \to (q \to p)
T T T T
T F T T
F T F T
F F T T

Tautology[edit]

Definition[edit]

A formula \theta \,\! of propositional logic is a tautology if only T's occur in the \theta \,\! column of the truth table.

Examples[edit]

Truth table for (p \to \lnot p) \to \lnot p = \theta

p \,\! \lnot p p \to \lnot p \theta \,\!
T F F T
F T T T

Truth table for (p \to (q \leftrightarrow \lnot q)) \to \lnot p = \theta

p \,\! q \,\! \lnot q q \leftrightarrow \lnot q p \to (q \leftrightarrow \lnot q) \lnot p \theta \,\!
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 (\lnot p \vee q) \to (p \to q) = \theta

p \,\! q \,\! \lnot p \lnot p \vee q p \to q \theta \,\!
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 \varphi \,\! and \theta \,\! are tautologically equivalent if \varphi \leftrightarrow \theta is a tautology.

Examples[edit]

Contraposition: p \to q = \theta is tautologically equivalent to \lnot q \to \lnot p = \varphi.

p \,\! q \,\! \lnot q \lnot p \theta \,\! \varphi \,\! \theta \leftrightarrow \varphi
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: \lnot (p \lor q) = \theta is tautologically equivalent to \lnot p \land \lnot q = \varphi.

p \,\! q \,\! \lnot p \lnot q p \lor q \theta \,\! \varphi \,\! \theta \leftrightarrow \varphi
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: \lnot (p \land q) = \theta is tautologically equivalent to \lnot p \lor \lnot q = \varphi. Truth table for Assignment #1

Related Resources[edit]

The materials in this course overlap with Introductory Discrete Mathematics for Computer Science, particularly Lesson 1.