Logiske ækvivalenser

Dette er en oversigt over logiske ækvivalenser, så de er samlet ét sted! Dog er nogle udsagn udladt, da de simpelthen er for simple.

Generelt sande udsagn

 * $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$
 * $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$
 * $$(A \cap B)^c = A^c \cup B^c$$
 * $$(A \cup B)^c = A^c \cap B^c$$
 * $$f(A \cup B) = f(A) \cup f(B)$$
 * $$f(A \cap B) \neq f(A) \cap f(B)$$
 * $$f(A \backslash B) \neq f(A) \backslash f(B)$$
 * $$(p \wedge q) \vee r \equiv (p \vee r) \wedge (q \vee r)$$
 * $$(p \vee q) \wedge r \equiv (p \wedge r) \vee (q \wedge r)$$
 * $$\neg (p \wedge q) \equiv \neg p \vee \neg q$$
 * $$\neg (p \vee q) \equiv \neg p \wedge \neg q$$
 * $$p \Rightarrow q \equiv \neg p \vee q$$
 * $$\neg (p \Rightarrow q) \equiv p \wedge \neg q$$
 * $$p \Rightarrow q \equiv \neg q \Rightarrow \neg p$$
 * $$p \vee q \equiv \neg p \Rightarrow q$$
 * $$p \Rightarrow (q \vee r) \equiv (p \wedge \neg q) \Rightarrow r$$
 * $$(\exists x\ \forall y : p) \Rightarrow (\forall y\ \exists x : p)$$
 * $$\neg (\forall x : p) \equiv \exists x: \neg p$$
 * $$\neg (\exists x : p) \equiv \forall x: \neg p$$