Logic and Truth Tables

These flashcards cover key concepts related to logic, truth tables, and various logical operations and properties.

Negation

The inverse of a proposition, typically denoted as ¬P.

Conjunction

A logical operator that combines two propositions, represented as P ∧ Q, which is true only when both propositions are true.

Disjunction

A logical operator that combines two propositions, represented as P ∨ Q, which is true when at least one proposition is true.

Exclusive Or (XOR)

A logical operation where the result is true if exactly one of the propositions is true, typically represented as P ⊕ Q.

Conditional Statement

A statement in the form P → Q, indicating that if P is true, then Q is also true.

Biconditional

A logical statement where both propositions are equivalent, denoted as P ↔ Q, meaning P is true if and only if Q is true.

Tautology

A proposition that is always true, regardless of the truth values of its components.

De Morgan's Laws

Laws that relate conjunctions and disjunctions through negation: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q and ¬(P ∨ Q) ≡ ¬P ∧ ¬Q.

Quantifiers

Expressions that indicate the quantity of specimens in a statement: universal (∀) and existential (∃).

Universal Instantiation

A rule stating that if a property is true for all elements, it is also true for a specific element.

Existential Generalization

A rule stating that if a property is true for some specific element, then it is true for at least one element in the domain.

Proof by Contradiction

A method of proving the truth of a statement by assuming its falsity and deriving a contradiction.

Proof by Contraposition

A method where one proves a conditional statement by proving that if the conclusion is false, then the hypothesis must also be false.

Logical Equivalence

Two statements that have the same truth values in all possible scenarios.