Propositional logic

$Logical Constant$

T : true F : false

Proposition: Statement that takes value

T

or

F

.

Compound proposition:

proposition that consists of two propositions joined by the

connective

\land, \lor, \neg

, \to

.

Tautology:

compound proposition that is true in every possible interpretation

A \Rightarrow B :

A \to B

is a tautology

A \Leftrightarrow B :

A \leftrightarrow B

is a tautology

\neg T \Leftrightarrow F

\neg F \Leftrightarrow T

P \land T \Leftrightarrow P

P \land F \Leftrightarrow F

P \lor T \Leftrightarrow T

P \lor F \Leftrightarrow P

\neg\neg P \Leftrightarrow P

P \land Q \Leftrightarrow Q \land P

P \lor Q \Leftrightarrow Q \lor P

(P \land Q) \land R \Leftrightarrow P \land (Q \land R)

(P \lor Q) \lor R \Leftrightarrow P \lor (Q \lor R)

P \land (Q \lor R) \Leftrightarrow (P \land Q) \lor (Q \land R)

P \lor (Q \land R) \Leftrightarrow (P \lor Q) \land (Q \lor R)

P \land (P \lor Q) \Leftrightarrow P

P \lor (P \land Q) \Leftrightarrow P

P \lor (\neg P) \Leftrightarrow T

P \land (\neg P) \Leftrightarrow F

P \land P \Leftrightarrow P

P \lor P \Leftrightarrow P

P \to Q \Leftrightarrow (\neg Q) \to (\neg P)

\neg (P \land Q) \neg (P \lor Q) \neg (P \lor Q) \neg (P_{1} \land P_{2} \land ... \land P_{n}) \Leftrightarrow \neg P \lor \neg Q \Leftrightarrow \neg P \land \neg Q \Leftrightarrow \neg P \land \neg Q \Leftrightarrow \neg P_{1} \lor \neg P_{2} \lor ... \lor \neg P_{n}