What are the basic theorems of Boolean algebra?

Answer 1

The Boolean prime ideal theorem:Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that and IF are disjoint. Then I is contained in some prime ideal of B that is disjoint from F. The consensus theorem:(X and Y) or ((not X) and Z) or (Y and Z) ≡ (X and Y) or ((not X) and Z)

xy + x'z + yz ≡ xy + x'z

De Morgan's laws:

NOT (P OR Q) ≡ (NOT P) AND (NOT Q)

NOT (P AND Q) ≡ (NOT P) OR (NOT Q)

AKA:

(P+Q)'≡P'Q'

(PQ)'≡P'+Q'

AKA:

¬(P U Q)≡¬P ∩ ¬Q

¬(P ∩ Q)≡¬P U ¬Q

Duality Principle:If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets. The laws of classical logicPeirce's law:((P→Q)→P)→P

P must be true if there is a proposition Q such that the truth of P follows from the truth of "if Pthen Q". In particular, when Q is taken to be a false formula, the law says that if P must be true whenever it implies the false, then P is true.

Stone's representation theorem for Boolean algebras:

Every Boolean algebra is isomorphic to a field of sets.

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