Central role of discrete mathematics in information technology and communications

Central role of discrete mathematics in information technology and communications

1. OBJECTIVE

   • Familiarize students with main mathematical concepts

2. Computer

   • finite structure= finite mathematics
   • Language of predicate logic
     • Logic & computers
       • studies notions, judments and reasoning as well as the laws of correct reasoning
         • formal language
           • formulas
         • Intended meaning
           • inferences
         • OBJETIVES
           • systematization of what we do daily
         • APLICATIONS
           • switching and logic networks, introduction to PROLOG , verification of correctness of computers programs
       • PREDICATE LOGIC
         • system of formal logic
           • connevtives AND, OR, NOT, IMPLIES, EQUIVALENT and QUANTIFIERS
             • associates a truth strcuture
               • predicates are statements about objects, their properties and relationship with other objects
                 • only things can be arguments
                 • predicates are terms
       • Fuzzy logic
         • generalizes the classical (two-valued) logic and includes many valued logic
           • make inferences from information that is imprecise, incomplete, uncertain or paryially true
             • imprecise knowledge
               • values of functions [ 0,1]
                 • indicates the degree of membership
         • theory of fuzzy subsets
           • generalization of the usual sets theory
             • contruction of membership
               • reflects the context within the concrete problem is being studied
         • Random
           • results from incertitude
             • shows the existence of varios degrees of belonging
         • allows implementing approximat reasoning
           • general knowledge
         • computationally simpler and faster than other
         • philosophical view
           • depends on the semantic interpretation of every fuzzy predicate
           • capacity to model linguistic knowledge
       • why do we need it ?
         • Showing that a problem can be solved by a computer program
         • translating problem description into a program language
         • arguing that a computer is correct and efficient
         • applying the new techniques of programming which require the mastery of different aspects of logic
     • Computer organization
       • Computer program viewed as a formula
         • has
           • Formal syntax, proof rules
   • Integrated circuits, transistors,resistors,capacitors, and other electronic components
     • transistors act as amplifiers or switches
       • switches are combined to form logic gates
       • bricks of implementation
   • Boolean alegebra

3. IMPORTANCE

   • Knowledge management,software engineering, programing languages/ data structures, hardware topics

4. GRAPHS & TREES

   • Neural nets
     • Interconnected assemblies of simple processing elements
       • vertices: placed on ordered layers
         • arrows: only between a vertex on layer l and a vertex on layer l+1, has weught (+ or -)
   • Semantic nets
     • vertices: represent objects
       • arrows: relationships between objects
   • Bayesian
     • acylic graphs
       • vertex: represent a random variable
         • arrow: conditional dependence
   • Minimum spanning trees
     • sub-graphs of weighted and connected graph
       • contain all vertices
   • Binary trees
     • mathematical & logical expressions
       • possibly involving parentheses
         • vertices : represent operators
           • leaves: represent values

5. References

   • Gersting,J., Mathematical Structures for Computer Science, Freeman& Cie,2003
   • Hein,J.L., Discrete Mathematucs, Jones and Bartlett Publishers, 1996.
   • Rosen,H.K,Discrete Mathematics,McGraw-Hill,2003
   • Reghis,M.Roventa.E,Clasical and Fuzzy Concepsts in Mathematical Logic and Applications, CRCPress 1998
   • Zadeh,L.A.,Probability Theory and Fuzzy Logic are Complementary rather than Competitive, Technometrics,August 1995, Vol.37,No.3
   • Zadeh,L.A.,Toward a Generalized Theory of Uncertainty(GTU)-An Outline,Information Sciences,Elsevier,Vol.172,pp.1-40,2005