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Introduction to Boolean Algebra

Introduction to Boolean Algebra. CSC 333. A nod to history . . . George Boole English mathematician Developed a mathematical model for logical thinking Mathematical Analysis of Logic ( 1847) Applicability of Boolean logic to electrical circuits independently recognized in writings by

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Introduction to Boolean Algebra

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  1. Introduction to Boolean Algebra CSC 333 CSC 333

  2. A nod to history . . . • George Boole • English mathematician • Developed a mathematical model for logical thinking • Mathematical Analysis of Logic (1847) • Applicability of Boolean logic to electrical circuits independently recognized in writings by • Claude Shannon (USA, 1937) • Victor Shestakov (Russia, 1941) CSC 333

  3. Boolean Algebra • Definition: A mathematical model of propositional logic and set theory. • Revisit the properties of wffs in chapter one and of sets in chapter three. • Note the similarities. CSC 333

  4. Boolean Algebra • “v” means disjunction. • “^” means conjunction. • “0” -> contradiction. • “1” -> tautology. • Propositional logic, set theory, and Boolean logic share common properties. CSC 333

  5. Boolean Algebra Structure • Similarities between propositional logic and set theory: • Both are concerned with sets • Set of wffs • Sets of subsets of a set • Both have 2 binary operations and 1 unary operation. • Each has 2 distinct elements. • Each has 10 properties. • These are distinguishing features of a Boolean algebra.

  6. Boolean Algebra defined . . . • Definition: A set B having • Defined binary operations + and ∙ • Defined unary operation ‘ • Two distinct elements 0 and 1 • With the following properties for elements of B: • Commutativity • Associativity • Distribution • Identity • Complementation CSC 333

  7. Example • B ={0, 1} • +, · and ‘ defined by truth tables, p. 538. • The properties defining a Boolean algebra can be demonstrated to hold for B. • Try practice 1, p. 538. CSC 333

  8. Proofs using Boolean Algebra • Note that in order to apply the properties of Boolean algebra, there must be an exact match of patterns. • See demonstration, p. 539. CSC 333

  9. Isomorphism: • The mapping of elements of one instance to the elements of the other so that crucial properties are preserved. • [Recall the definition of bijection: If every element in the domain has a unique image in the codomain and every element in the codomain has a unique preimage, then the function is a one-to-one onto function, also known as a bijective function, or a bijection.] • See Example 5, p. 542. • See definition, p. 544. CSC 333

  10. Theorem on Finite Boolean Algebras • The number of elements in a Boolean algebra, B, is a power of 2 (say, 2m). • B is isomorphic to a power set with elements from 1 to m. CSC 333

  11. Summary • Boolean algebras, propositional logic, and set theory are abstractions of common properties. • If an isomorphism exists from A to B, which are instances of a structure, then A and B are equivalent except for labels. • All finite Boolean algebras are isomorphic to Boolean algebras that are power sets. CSC 333

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