introduction to boolean algebra
Download
Skip this Video
Download Presentation
Introduction to Boolean Algebra

Loading in 2 Seconds...

play fullscreen
1 / 11

Introduction to Boolean Algebra - PowerPoint PPT Presentation


  • 105 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Introduction to Boolean Algebra' - britain


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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

ad