Logical Equivalence Laws

If two statements are logically equivalent, then their Truth Tables are the same.

De Morgan’s Laws

$\neg (P\wedge Q)\equiv \neg P\vee \neg Q$

$$\neg (P\vee Q)\equiv \neg P\wedge \neg Q$$

$\overline{A\cup B}=\overline{A}\cap \overline{B}$

Commutative Laws

$P\wedge Q\equiv Q\wedge P$

$$P\vee Q\equiv Q\vee P$$

Associative Laws

$$P\wedge (Q\wedge R)\equiv (P\wedge Q)\wedge R$$ $$P\vee (Q\vee R)\equiv (P\vee V)\vee R$$

Idempotent Laws

$P\wedge P\equiv P$

$$P\vee P\equiv P$$

Distributive Laws

$$P\wedge (Q\vee R)\equiv (P\wedge Q)\vee (P\wedge R)$$ $$P\vee (Q\wedge R)\equiv (P\vee Q)\wedge (P\vee R)$$

Absorption Laws

$$P\vee (P\wedge Q)\equiv P$$ $$P\wedge (P\vee Q)\equiv P$$

Double Negation Law

$$\neg \neg P\equiv P$$

Conditional Laws

$P⟹Q\equiv \neg P\vee Q$

$$P⟹Q\equiv \neg (P\wedge \neg Q)$$

Contrapositive Law

$P⟹Q\equiv \neg Q⟹\neg P$

Biconditional Law

$P⟺Q\equiv P⟹Q\wedge Q⟹P$

Domination Laws

Derived from Tautology and Contradiction. $P\vee T\equiv T$ $P\wedge F\equiv F$

Quantifier Exchange

$\neg (\forall x,P(x))\equiv \exists x,\neg P(x)$ $\neg (\exists x,P(x))\equiv \forall x,\neg P(x)$