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In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations. Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if: a ¤ ( a ⁂ b ) = a ⁂ ( a ¤ b ) = a .
The absorption rule may be expressed as a sequent: ()where is a metalogical symbol meaning that () is a syntactic consequence of () in some logical system; . and expressed as a truth-functional tautology or theorem of propositional logic.
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.
All concrete Boolean algebras satisfy the laws (by proof rather than fiat), whence every concrete Boolean algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set and certain operations satisfying certain laws or axioms by fiat is entirely analogous to the abstract definitions of group ...
Boolean algebra is intimately related to propositional logic ... Absorption law; ... Complete Boolean algebra; Consensus theorem;
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 algebra; Truth table; ... and each equivalence between logical forms must be either an axiom or provable as a theorem. ... ∨ satisfies the absorption law ...
De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.