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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.
1. A boolean algebra is a complemented distributive lattice. (def) 2. A boolean algebra is a heyting algebra. [1] 3. A boolean algebra is orthocomplemented. [2] 4. A distributive orthocomplemented lattice is orthomodular. 5. A boolean algebra is orthomodular. (1,3,4) 6. An orthomodular lattice is orthocomplemented. (def) 7. An orthocomplemented ...
The Hasse diagram of the free Boolean algebra on two generators, p and q. Take p (left circle) to be "John is tall" and q (right circle)to be "Mary is rich". The atoms are the four elements in the row just above FALSE. The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John ...
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 ...
A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced the technique in 1953 [1] [2] as a refinement of Edward W. Veitch's 1952 Veitch chart, [3] [4] which itself was a rediscovery of Allan Marquand's 1881 logical diagram [5] [6], or Marquand diagram. [4])
The representation of R div as a Boolean matrix is shown in the middle table; the representation both as a Hasse diagram and as a directed graph is shown in the left picture. The following are equivalent: x R div y is true. (x,y) ∈ R div. A path from x to y exists in the Hasse diagram representing R div.