Docstoc

Discrete Mathematics Lecture

Document Sample
Discrete Mathematics Lecture Powered By Docstoc
					  Discrete Structures
   Chapter 1 Part B
Fundamentals of Logic
 Nurul Amelina Nasharuddin
  Multimedia Department


                             1
                Predicates
• A predicate is a sentence that contains a
  finite number of variables and becomes a
  statement when specific values are
  substituted for the variables
• The domain of a predicate variable is a set
  of all values that may be substituted in place
  of the variable
• P(x): x is a student at UPM
• P(x, y): x is a student at y
                                               2
                              Predicates
• The sets in which predicate variables take their values may
  be described either in words or in symbols
• x  A indicates that x is an element of the set A
• x  A; x is not an element of the set A
• {1, 2, 3} refers to the set whose elements are 1, 2 and 3
• Certain set of numbers that frequently referred to are
       Symbol                         Set                          Example

         R      Set of all real numbers                    All points on a line number

         Z      Set of all integers                        …,-3, -2, -1, 0, 1, 2, 3, …

         Q      Set of all rational numbers, or quotient   {p/q | p ∈ Z, q ∈ Z, q ≠ 0}
                of integers
                                                                                         3
                      Predicates
• If P(x) is a predicate and x has domain D, the truth set of
  P(x) is the set of all elements in D that make P(x) true
  when substituted for x. The truth set is denoted as:
                            {x  D | P(x)}
   which is read “the set of all x in D such that P(x)”
• True or false?: Let P(x): x2 > x with domain set R (real
  numbers).




                                                                4
                     Predicates
• Let Q(n) be the predicate “n is a factor of
  8.” Find the truth set of Q(n) if
   – the domain of n is the set of Z+
   – the domain of n is the set of Z

• Ans:
   •{1, 2, 4, 8}
   •{-1, -2, -4, -8, 1, 2, 4, 8}

                                                5
                  Predicates
• Let P(x) and Q(x) be predicates with the common
  domain D
• P(x)  Q(x) means that every element in the truth
  set of P(x) is in the truth set of Q(x), or
  equivalently for every x, P(x)  Q(x)
• P(x)  Q(x) means that P(x) and Q(x) have
  identical truth sets, or equivalently for every x,
  P(x)  Q(x)

                                                   6
        Universal Quantifier, 
• Let P(x) be a predicate with domain D.
• A universal statement is a statement in the form
  “x  D, P(x)”.  denotes “for all”
• It is true iff P(x) is true for every x from D.
  It is false iff P(x) is false for at least one x from D.
• A value of x for which P(x) is false is called a
  counterexample to the universal statement
• Examples
   – D = {1, 2, 3, 4, 5}: x  D, x² >= x        (True)
   – x  R, x² >= x                             (False)
• Method of exhaustion                                     7
        Existential Quantifier, 
• Let P(x) be a predicate with domain D.
• An existential statement is a statement in the form
  “x  D, P(x)”.  denotes “there exists”
• It is true iff P(x) is true for at least one x from D.
  It is false iff P(x) is false for every x from D.
• Examples:
   – m  Z, m² = m                             (True)
   – E = {5, 6, 7, 8, 9}, x  E, m² = m        (False)


                                                           8
                   Quantifier
• Rewrite the following without using  and
  :
  a. x  R, x2  0
  b. x  Z such that m2 = m
• Ans:
  a. All real numbers have nonnegative squares
     The square of any real number is nonnegative
  b. m2 =m, for some integer m
     Some integers equal their own square
                                                    9
Universal Conditional Statement
• Universal conditional statement
               “x, if P(x) then Q(x)”:
• Eg: x  R, if x > 2, then x2 > 4
• Universal conditional statement is called
  vacuously true or true by default iff P(x) is false
  for every x in D
• Eg:
  When x =1, if 1 > 2, then 12 > 4             (True)
  When x = -3, if (-3) > 2, then (-3)2 > 4     (True)
                                                        10
       Negation of Quantified
            Statements
• The negation of a universally quantified
  statement x  D, P(x) is x  D, ~P(x)
    (x  D, P(x))  x  D such that ~P(x)
• The negation of an existentially quantified
  statement x  D, P(x) is x  D, ~P(x)
  ( x  D, P(x))   x  D such that ~P(x)
• Rewrite formally, formal and informal
  negation: No politicians are honest
                                               11
                  Answer
No politicians are honest
• Formal version:  politicians x, x is not
  honest
• Formal negation:  a politician x such that x
  is honest
• Informal negation: Some politicians are
  honest

                                              12
        Negation of Quantified
             Statements
• The negation of a universal conditional
  statement x  D, P(x)  Q(x) is x  D,
  P(x)  ~Q(x)
  (x, if P(x) then Q(x))  x such that P(x) and ~Q(x)
• Noted before; ~(p  q)  p  ~q
• Write formal negation:  people p, of p is
  blond then p has blue eyes
• Ans:  a person such that p is blond and p
  does not have blue eyes
                                                           13
                            Exercises
• Write negations for each of the following statements:
   –   All dinosaurs are extinct
   –   No irrational numbers are integers
   –   Some exercises have answers
   –   All COBOL programs have at least 20 lines
   –   The sum of any two even integers is even
   –   The square of any even integer is even




                                                          14
 Multiply Quantified Statements
• Imagine you are in a factory and the guide tell you
   “There is person supervising every detail of the
                   production process”
• Have both existential quantifier There is and
  universal quantifier every
• When a statement contains more than one
  quantifier, the actions being performed is in the
  order in which the quantifier occur.

                                                    15
 Multiply Quantified Statements
• To establish the truth of the statement “x in D,
  y in E such P(x, y)”
   Pick whatever x from set D , then find element
     y in E that “works” for that particular y in P(x,
     y)
• To establish the truth of the statement “x in D,
  such that y in E, P(x, y)”
   Find element x in D that “works” no matter y in
     P(x, y)
                                                     16
 Multiply Quantified Statements
• For all positive numbers x, there exists number y
  such that y < x
• There exists number x such that for all positive
  numbers y, y < x
• For all people x there exists person y such that x
  loves y (Informally: Everybody loves somebody)
• There exists person x such that for all people y, x
  loves y (Informally: Somebody loves everybody)


                                                        17
Negation of Multiply Quantified
          Statements
• The negation of x, y, P(x, y)
  is logically equivalent to x, y, ~P(x, y)
• The negation of x, y, P(x, y)
  is logically equivalent to x, y, ~P(x, y)




                                                18
Negation of Multiply Quantified
          Statements
• Rewrite formally and negate: Everybody
  trusts somebody.
• Formally: (a)  people x,  a person y such
  that x trusts y.
• Negation: Somebody trusts nobody.
• Formally: (b)  a person x, such that 
  people y, x does not trust y.

                                            19
      Necessary and Sufficient
        Conditions, Only If
• x, r(x) is a sufficient condition for s(x)
  means: x, if r(x) then s(x)
• x, r(x) is a necessary condition for s(x)
  means: x, if s(x) then r(x)
• x, r(x) only if s(x) means: x, if r(x) then
  s(x)


                                                  20
Logical Equivalence and Logical
  Implication for Quantified
  Statements in One Variable
x  D, (P(x)  Q(x)  (x  D, P(x))  (x  D, Q(x))
 x  D, (P(x)  Q(x))  (x  D, P(x))  (x  D, Q(x))
(x  D, P(x))  (x  D, Q(x))  x  D, (P(x)  Q(x))
 x  D, (P(x)  Q(x))  (x  D, P(x))  (x  D, Q(x))



                                                           21
      Application: Digital Logic
               Circuits
• Digital Logic Circuit is a basic electronic component of a
  digital system
• Values of digital signals are 0 or 1 (bits)
• Black Box is specified by the signal input/output table
• Input/output table = truth table
• Three gates: AND-gate, OR-gate, NOT-gate
     Application: Digital Logic
              Circuits
• Combinational circuit is a combination of
  logical gates
• Combinational circuit always correspond to
  some Boolean expression, such that
  input/output table of a table and a truth table
  of the expression are identical


                                               23
     Application: Digital Logic
              Circuits
• Finding Boolean expression for a circuit:
  P  ~ (Q  R)




                                              24
     Application: Digital Logic
              Circuits
• Determining output for a given input: Move
  from left to right through the diagram,
  tracing the action of each gate on the input
  signals
• Constructing the input/output table for a
  circuit: list the all possible combinations for
  input signals, and find the output for each
  by tracing through the unit
                                                25
      Application: Digital Logic
               Circuits
• A recognizer is a circuit that outputs 1 for exactly
  one particular combination of input signals and
  outputs 0’s for all other combinations
• Multiple-input AND and OR gates
• Finding a circuit that corresponds to a given
  input/output table:
   – Construct equivalent Boolean expression using
     disjunctive normal form: for all outputs of 1 construct a
     conjunctive form based on the truth table row. All
     conjunctive forms are united using disjunction
   – Construct a digital logic circuit equivalent to the
     Boolean expression
      Application: Digital Logic
               Circuits
• Given                     • Boolean expression:
                            (P  Q  R) 
       Input       Output

  P     Q      R     S

  1     1      1     1      (P  Q  R) 
  1     1      0     0
                            (P  Q  R)
  1     0      1     1

  1     0      0     1      • Circuit:
  0     1      1     0

  0     1      0     0

  0     0      1     0

  0     0      0     0
                                                    27
     Application: Digital Logic
              Circuits
• Two digital logic circuits are equivalent iff
  their input/output tables are identical
• Simplification of circuits by using Theorem
  1.1.1 to simplify the Boolean expression
• Another way to simplify circuit by using
  – Scheffer stroke, | (NAND, AND-gate followed
    by NOT-gate)
  – Peirce arrow,  (NOR, OR-gate followed by
    NOT-gate)
                 Exercises
• Construct circuits for (P  Q)  R
• From the given input/output        Input       Output

                                  P    Q     R     S
  table below, construct
                                  1    1     1     0

  (a) a Boolean expression        1    1     0     1

                                  1    0     1     0
  (b) a circuit                   1    0     0     0

                                   0    1    1     1

                                   0    1    0     0

                                   0    0    1     0

                                   0    0    0     0

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:7
posted:8/7/2011
language:English
pages:29