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Mathematical Logic Formulas

Idempotent Laws
Commutative Laws
Associative Laws
Distributive Laws
Identity Laws
Domination Laws
Negation Laws
Double Negation
De Morgan Laws
Absorption Laws
Redundancy Laws
Monotonicity Laws
Conditional Equivalences
Biconditional Equivalences
Tautology and Contradiction Duality
32 formulas

Idempotent Laws

(2 formulas)

Idempotent Law for Conjunction

P∧P≡PP \land P \equiv P
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Idempotent Law for Disjunction

P∨P≡PP \lor P \equiv P
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Commutative Laws

(2 formulas)

Commutative Law for Conjunction

P∧Q≡Q∧PP \land Q \equiv Q \land P
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Commutative Law for Disjunction

P∨Q≡Q∨PP \lor Q \equiv Q \lor P
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Associative Laws

(2 formulas)

Associative Law for Conjunction

(P∧Q)∧R≡P∧(Q∧R)(P \land Q) \land R \equiv P \land (Q \land R)
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Associative Law for Disjunction

(P∨Q)∨R≡P∨(Q∨R)(P \lor Q) \lor R \equiv P \lor (Q \lor R)
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Distributive Laws

(2 formulas)

Distributive Law of Conjunction over Disjunction

P∧(Q∨R)≡(P∧Q)∨(P∧R)P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)
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Distributive Law of Disjunction over Conjunction

P∨(Q∧R)≡(P∨Q)∧(P∨R)P \lor (Q \land R) \equiv (P \lor Q) \land (P \lor R)
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Identity Laws

(2 formulas)

Identity Law for Conjunction

P∧⊤≡PP \land \top \equiv P
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Identity Law for Disjunction

P∨⊥≡PP \lor \bot \equiv P
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Domination Laws

(2 formulas)

Domination Law for Conjunction

P∧⊥≡⊥P \land \bot \equiv \bot
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Domination Law for Disjunction

P∨⊤≡⊤P \lor \top \equiv \top
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Negation Laws

(2 formulas)

Law of Excluded Middle

P∨¬P≡⊤P \lor \neg P \equiv \top
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Law of Non Contradiction

P∧¬P≡⊥P \land \neg P \equiv \bot
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Double Negation

(1 formula)

Double Negation Law

¬(¬P)≡P\neg(\neg P) \equiv P
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De Morgan Laws

(2 formulas)

De Morgan Law for Conjunction

¬(P∧Q)≡¬P∨¬Q\neg(P \land Q) \equiv \neg P \lor \neg Q
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De Morgan Law for Disjunction

¬(P∨Q)≡¬P∧¬Q\neg(P \lor Q) \equiv \neg P \land \neg Q
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Absorption Laws

(2 formulas)

Absorption Conjunction Form

P∧(P∨Q)≡PP \land (P \lor Q) \equiv P
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Absorption Disjunction Form

P∨(P∧Q)≡PP \lor (P \land Q) \equiv P
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Redundancy Laws

(2 formulas)

Redundancy Law for Disjunction

P∨(Q∨P)≡P∨QP \lor (Q \lor P) \equiv P \lor Q
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Redundancy Law for Conjunction

P∧(Q∧P)≡P∧QP \land (Q \land P) \equiv P \land Q
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Monotonicity Laws

(2 formulas)

Disjunction Introduction

P→(P∨Q)P \to (P \lor Q)
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Conjunction Elimination

(P∧Q)→P(P \land Q) \to P
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Conditional Equivalences

(4 formulas)

Material Implication

P→Q≡¬P∨QP \to Q \equiv \neg P \lor Q
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Contrapositive Equivalence

P→Q≡¬Q→¬PP \to Q \equiv \neg Q \to \neg P
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Negation of a Conditional

¬(P→Q)≡P∧¬Q\neg(P \to Q) \equiv P \land \neg Q
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Exportation

(P∧Q)→R≡P→(Q→R)(P \land Q) \to R \equiv P \to (Q \to R)
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Biconditional Equivalences

(3 formulas)

Biconditional as Two Conditionals

P↔Q≡(P→Q)∧(Q→P)P \leftrightarrow Q \equiv (P \to Q) \land (Q \to P)
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Biconditional as Disjunction of Conjunctions

P↔Q≡(P∧Q)∨(¬P∧¬Q)P \leftrightarrow Q \equiv (P \land Q) \lor (\neg P \land \neg Q)
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Negation of a Biconditional

¬(P↔Q)≡(P∧¬Q)∨(¬P∧Q)\neg(P \leftrightarrow Q) \equiv (P \land \neg Q) \lor (\neg P \land Q)
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Tautology and Contradiction Duality

(2 formulas)

Negation of Tautology

¬⊤≡⊥\neg \top \equiv \bot
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Negation of Contradiction

¬⊥≡⊤\neg \bot \equiv \top
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Idempotent Laws
Idempotent Law for ConjunctionIdempotent Law for Disjunction
Commutative Laws
Commutative Law for ConjunctionCommutative Law for Disjunction
Associative Laws
Associative Law for ConjunctionAssociative Law for Disjunction
Distributive Laws
Distributive Law of Conjunction over DisjunctionDistributive Law of Disjunction over Conjunction
Identity Laws
Identity Law for ConjunctionIdentity Law for Disjunction
Domination Laws
Domination Law for ConjunctionDomination Law for Disjunction
Negation Laws
Law of Excluded MiddleLaw of Non Contradiction
Double Negation
Double Negation Law
De Morgan Laws
De Morgan Law for ConjunctionDe Morgan Law for Disjunction
Absorption Laws
Absorption Conjunction FormAbsorption Disjunction Form
Redundancy Laws
Redundancy Law for DisjunctionRedundancy Law for Conjunction
Monotonicity Laws
Disjunction IntroductionConjunction Elimination
Conditional Equivalences
Material ImplicationContrapositive EquivalenceNegation of a ConditionalExportation
Biconditional Equivalences
Biconditional as Two ConditionalsBiconditional as Disjunction of ConjunctionsNegation of a Biconditional
Tautology and Contradiction Duality
Negation of TautologyNegation of Contradiction