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

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Introduction to boolean algebra

Introduction to Boolean Algebra

CSC 333

CSC 333


A nod to history

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

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 algebra1

Boolean Algebra

  • “v” means disjunction.

  • “^” means conjunction.

  • “0” -> contradiction.

  • “1” -> tautology.

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

CSC 333


Boolean algebra structure

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

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

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


Proofs using boolean algebra

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


Isomorphism

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


Theorem on finite boolean algebras

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


Summary

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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