Introduction to Boolean Algebra

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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

CSC 333

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
• Claude Shannon (USA, 1937)
• Victor Shestakov (Russia, 1941)

CSC 333

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

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

CSC 333

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.
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

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.

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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.

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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.

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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.

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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