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A Boolean algebra is a set A, equipped with two binary operations ∧ (called "meet" or "and"), ∨ (called "join" or "or"), a unary operation ¬ (called "complement" or "not") and two elements 0 and 1 in A (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols ⊥ and ā¤, respectively), such that for all elements a, b and c of A, the following axioms hold: [2]
A law of Boolean algebra is an identity such as x ∨ (y ∨ z) = (x ∨ y) ∨ z between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡ ...
The complement of the intersection of two sets is the same as the union of their complements; or not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B. De Morgan's law with set subtraction operation
In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that B = {0, 1}. Paul Halmos's name for this algebra "2" has some following in the literature, and will be employed here.
The authors demonstrate a proof that any Boolean (logic) function can be expressed in either disjunctive or conjunctive normal form (cf pages 5–6); the proof simply proceeds by creating all 2 N rows of N Boolean variables and demonstrates that each row ("minterm" or "maxterm") has a unique Boolean expression. Any Boolean function of the N ...
Corner quotes, also called “Quine quotes”; for quasi-quotation, i.e. quoting specific context of unspecified (“variable”) expressions; [3] also used for denoting Gödel number; [4] for example “āGā” denotes the Gödel number of G. (Typographical note: although the quotes appears as a “pair” in unicode (231C and 231D), they ...
In Boolean functions, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal. For example, if A {\displaystyle A} , B {\displaystyle B} and C {\displaystyle C} are variables then the expression A ¯ B C {\displaystyle {\bar {A}}BC} contains three literals and the expression A ¯ C + B ¯ C ...
Within an expression containing two or more of the same associative connectives in a row, the order of the operations does not matter as long as the sequence of the operands is not changed. Commutativity The operands of the connective may be swapped, preserving logical equivalence to the original expression. Distributivity