Branch of logic which studies the way in which propositions, statements which are true or false, can be combined, reduced, and modified. Any proposition can be analyzed using a truth table, ******which analyzes the truth of a propositional form based on the truth of its components.
Any propositional form which is always true is said to be a tautology and any propositional form which is always false, or impossible to be true, is said to be a contradiction.
Operators which can be used to join or alter propositions to form propositional forms.
Binary operators which act like AND and OR gates between two propositions.
Conjunction. $P \wedge Q$ is true if both $P$ and $Q$ are true.
Disjunction. $P \vee Q$ is true if $P$ is true, $Q$ is true, or if both are true.
Both operators are associative and distributive over the other.
$$ (P\vee Q)\wedge(S\vee R) \equiv (P\wedge S) \vee (P\wedge R) \vee (Q\wedge S) \vee (Q \wedge R) \\ (P\wedge Q)\vee (S\wedge R) \equiv (P\vee S)\wedge (P \vee R) \wedge (Q \vee S) \wedge (Q\vee R) $$
Unary operator which forms a proposition which is true only when the operand is not.
Negation. $\neg P$ is true if $P$ is false.
The interaction between negation and other logical connectives is governed by De Morgan’s Law.
$$ \neg(P\wedge Q) \equiv \neg P\vee \neg Q \\ \neg(P \vee Q) \equiv \neg P \wedge \neg Q $$