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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 ⊤, 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 ≡ ...
This is a list of topics around Boolean algebra and ... Minimal axioms for Boolean algebra; ... Stone's representation theorem for Boolean algebras; People ...
A not too well known application of the Boolean prime ideal theorem is the existence of a non-measurable set [3] (the example usually given is the Vitali set, which requires the axiom of choice). From this and the fact that the BPI is strictly weaker than the axiom of choice, it follows that the existence of non-measurable sets is strictly ...
This is a list of axioms as that term is understood in mathematics. In epistemology , the word axiom is understood differently; see axiom and self-evidence . Individual axioms are almost always part of a larger axiomatic system .
Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.' [1] Just as group theory deals with groups, and linear algebra with vector spaces, Boolean algebras are models of the ...
Boolean meet and the constants 0 and 1 are easily defined from the Robbins algebra primitives. Pending verification of the conjecture, the system of Robbins was called "Robbins algebra." Verifying the Robbins conjecture required proving Huntington's equation, or some other axiomatization of a Boolean algebra, as theorems of a Robbins algebra.
Bonnet theorem (differential geometry) Boolean prime ideal theorem (mathematical logic) Borel–Bott–Weil theorem (representation theory) Borel–Carathéodory theorem (complex analysis) Borel–Weil theorem (representation theory) Borel determinacy theorem ; Borel fixed-point theorem (algebraic geometry) Borsuk–Ulam theorem