General logical equivalences

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Equivalence	Name
${\displaystyle p\wedge \top \equiv p}$ ${\displaystyle p\vee \bot \equiv p}$	Identity laws
${\displaystyle p\vee \top \equiv \top }$ ${\displaystyle p\wedge \bot \equiv \bot }$	Domination laws
${\displaystyle p\vee p\equiv p}$ ${\displaystyle p\wedge p\equiv p}$	Idempotent or tautology laws
${\displaystyle \neg (\neg p)\equiv p}$	Double negation law
${\displaystyle p\vee q\equiv q\vee p}$ ${\displaystyle p\wedge q\equiv q\wedge p}$	Commutative laws
${\displaystyle (p\vee q)\vee r\equiv p\vee (q\vee r)}$ ${\displaystyle (p\wedge q)\wedge r\equiv p\wedge (q\wedge r)}$	Associative laws
${\displaystyle p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)}$ ${\displaystyle p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)}$	Distributive laws
${\displaystyle \neg (p\wedge q)\equiv \neg p\vee \neg q}$ ${\displaystyle \neg (p\vee q)\equiv \neg p\wedge \neg q}$	De Morgan's laws
${\displaystyle p\vee (p\wedge q)\equiv p}$ ${\displaystyle p\wedge (p\vee q)\equiv p}$	Absorption laws
${\displaystyle p\vee \neg p\equiv \top }$ ${\displaystyle p\wedge \neg p\equiv \bot }$	Negation laws

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Logical equivalences involving conditional statements

1. ${\displaystyle p\implies q\equiv \neg p\vee q}$
2. ${\displaystyle p\implies q\equiv \neg q\implies \neg p}$
3. ${\displaystyle p\vee q\equiv \neg p\implies q}$
4. ${\displaystyle p\wedge q\equiv \neg (p\implies \neg q)}$
5. ${\displaystyle \neg (p\implies q)\equiv p\wedge \neg q}$
6. ${\displaystyle (p\implies q)\wedge (p\implies r)\equiv p\implies (q\wedge r)}$
7. ${\displaystyle (p\implies q)\vee (p\implies r)\equiv p\implies (q\vee r)}$
8. ${\displaystyle (p\implies r)\wedge (q\implies r)\equiv (p\vee q)\implies r}$
9. ${\displaystyle (p\implies r)\vee (q\implies r)\equiv (p\wedge q)\implies r}$

Logical equivalences involving biconditionals

1. ${\displaystyle p\iff q\equiv (p\implies q)\wedge (q\implies p)}$
2. ${\displaystyle p\iff q\equiv \neg p\iff \neg q}$
3. ${\displaystyle p\iff q\equiv (p\wedge q)\vee (\neg p\wedge \neg q)}$
4. ${\displaystyle \neg (p\iff q)\equiv p\iff \neg q\equiv p\oplus q}$

Where ${\displaystyle \oplus }$ represents XOR.

In logic

The following statements are logically equivalent:

1. If Lisa is in Denmark, then she is in Europe (a statement of the form ${\displaystyle d\implies e}$).
2. If Lisa is not in Europe, then she is not in Denmark (a statement of the form ${\displaystyle \neg e\implies \neg d}$).

Syntactically, (1) and (2) are derivable from each other via the rules of contraposition and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in Denmark is false or Lisa is in Europe is true.

(Note that in this example, classical logic is assumed. Some non-classical logics do not deem (1) and (2) to be logically equivalent.)