Logic

Notation

¬ - Negation, often people also use "!"

∧ - Conjunction (AND)

∨ - Disjunction (OR)

→ - Implication

↔ - Proposition

⊗ - Exclusive Or

Basics

x = ¬(¬x) this is double negation

x ∧ y = y ∧ x

x ∨ y = y ∨ x

z ∧ (x ∧ y) = (z ∧ x) ∧ y)

z ∨ (x ∨ y) = (z ∨ x) ∨ y)

Note #1: A statment is a tautology if its value always remains true. Example:

x ∨ ¬x = T

Note #2: A statment is a contradiction if its value always remains false. Example:

x ∧ ¬x = F

De Morgan's Laws

(¬x) ∧ (¬y) = ¬(x ∨ y)

(¬x) ∨ (¬y) = ¬(x ∧ y)

Implication

(a → b) ↔ ¬a ∨ b

(a → b) ↔ (¬b → ¬a)

(a → b) ↔ ¬(a ∧ ¬b)

Inference

((a → b)∧a)→ b - Modus Pones

((a → b)∧¬b)→ ¬a - Modus Tollens

(a → b)∧(b→c) → (a→c) - Hypothetical Syllogism

((a ∨ b)∧ ¬a) → b - Disjunctive Syllogism

Logic Tables