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Introduction to Boolean Algebra - PowerPoint PPT Presentation

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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• 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)

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• 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.

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• “v” means disjunction.

• “^” means conjunction.

• “1” -> tautology.

• Propositional logic, set theory, and Boolean logic share common properties.

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• 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.

• 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

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• 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.

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• Note that in order to apply the properties of Boolean algebra, there must be an exact match of patterns.

• See demonstration, p. 539.

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• 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.

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• 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.

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• 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.

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