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Propositional Equivalences

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Biconditional

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pq(pq)(qp)p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p)

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Double Negation Law

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¬(¬p)p\neg (\neg p) \equiv p

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Negation Laws

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p¬pTp \lor \neg p \equiv T and p¬pFp \land \neg p \equiv F

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Identity Laws

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pTpp \land T \equiv p and pFpp \lor F \equiv p

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Domination Laws

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pTTp \lor T \equiv T and pFFp \land F \equiv F

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Idempotent Laws

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pppp \land p \equiv p and pppp \lor p \equiv p

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De Morgan's Laws

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¬(pq)¬p¬q\neg (p \land q) \equiv \neg p \lor \neg q and ¬(pq)¬p¬q\neg (p \lor q) \equiv \neg p \land \neg q

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Implication

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pq¬pqp \rightarrow q \equiv \neg p \lor q

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Commutative Laws

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pqqpp \land q \equiv q \land p and pqqpp \lor q \equiv q \lor p

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Associative Laws

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(pq)rp(qr)(p \land q) \land r \equiv p \land (q \land r) and (pq)rp(qr)(p \lor q) \lor r \equiv p \lor (q \lor r)

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Distributive Laws

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p(qr)(pq)(pr)p \land (q \lor r) \equiv (p \land q) \lor (p \land r) and p(qr)(pq)(pr)p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)

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Absorption Laws

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p(pq)pp \land (p \lor q) \equiv p and p(pq)pp \lor (p \land q) \equiv p

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