Equivalences

Tags: #◇Problem Solving #◇Equivalences #◇CCDSTRU #◇Logic #◇M1

---

◇Important Formulas

Logical Equivalence Rules

Compound propositions whose truth values are the same for all possible cases are said to be logically equivalent ($p \equiv q$ or $p \Leftrightarrow q$).

Identity Laws: $p \land T \equiv p$ $p \lor F \equiv p$

Domination Laws: $p \land F \equiv F$ $p \lor T \equiv T$

Negation Laws: $p \land \neg p \equiv F$ $p \lor \neg p \equiv T$

Idempotent Laws: $p \land p \equiv p$ $p \lor p \equiv p$

Involution Law: $\neg(\neg p) \equiv p$

De Morgan's Laws: $\neg(p \land q) \equiv \neg p \lor \neg q$ $\neg(p \lor q) \equiv \neg p \land \neg q$

Commutative Laws: $p \land q \equiv q \land p$ $p \lor q \equiv q \lor p$

Associative Laws: $p \land (q \land r) \equiv (p \land q) \land r$ $p \lor (q \lor r) \equiv (p \lor q) \lor r$

Distributive Laws: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$ $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$

Absorption Laws: $p \land (p \lor q) \equiv p$ $p \lor (p \land q) \equiv p$

Material Implication Laws: $p \rightarrow q \equiv \neg p \lor q$

Material Equivalence Laws: $p \leftrightarrow q \equiv (p \rightarrow q) \land (q \rightarrow p)$ $p \leftrightarrow q \equiv (p \land q) \lor (\neg p \land \neg q)$

◇Worked Examples

Definitions of Truth Value Types Tautology (Fact): A compound proposition whose truth value is always true. Contradiction (Fallacy): A compound proposition whose truth value is always false. Contingency: A compound proposition whose truth value is not a tautology nor a contradiction.

[!tip] Verification
You can verify equivalence pairs (e.g., $p \rightarrow q$ and $\neg q \rightarrow \neg p$) using Rules of Logical Equivalences or by constructing truth tables and verifying that their outputs match for all possible truth assignments of their variables.

◇References & Navigation

• Related Notes: ◇Propositional Logic