CSC3190 Course Info by h7pc3hf

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									          Lecture 3: Logic (2)

   Propositional Equivalences
   Predicates and Quantifiers




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3.1. Propositional Equivalences

   Categories of compound propositions:
       • A tautology is a proposition which is always
         true.
          • Classic Example: PP
       • A contradiction is a proposition which is
         always false.
          • Classic Example: PP
       • A contingency is a proposition which neither a
         tautology nor a contradiction.
          • Example: (PQ)R
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3.1. Propositional Equivalences

   Two propositions P and Q are logically
    equivalent if P  Q is a tautology. We
    write
                  PQ




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3.1. Propositional Equivalences

   Example: (PQ)(QP)  (P  Q)
   Proof:
       • Left side and the right side must have the
         same truth values, independent of the
         truth value of the component propositions.
       • To show a proposition is not a tautology:
         use an abbreviated truth table
         • try to find a counter example or to disprove the
           assertion.
         • search for a case where the proposition is false.

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        3.1. Propositional Equivalences
                   • Two possible cases:
P Q      PQ
                      •   Case 1: Left side false, right side true.
0   0          1
                      •   Case 2: Left side true, right side false.
0   1          1
1
1
    0
    1
               0
               1
                   • Case 1: Try left side false, right side true
                      •   Left side false: only one of PQ or QP need be false.
                            1a. Assume PQ = F. Then P = T, Q = F. But then
P   Q     PQ                  right side PQ = F. Oops, wrong guess.
0   0          1            1b. Try QP = F. Then Q = T, P = F. But then right
0   1          0
                               side PQ = F. Another wrong guess.
1   0          0
1   1          1
                            *Proof for (PQ)(QP)  (P  Q)


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        3.1. Propositional Equivalences
                   • Case 2. Try left side true, right side false
P   Q     PQ
                         •   If right side is false, P and Q cannot have the same truth value.
0   0          1
                               2a. Assume P =T, Q = F. Then PQ = F and the conjunction
0   1          1
                                 must be false so the left side cannot be true in this case.
1   0          0
                                 Another wrong guess.
1   1          1
                               2b. Assume Q = T, P = F. Then QP = F. Again the left side
                                 cannot be true.
P   Q     PQ      • We have exhausted all possibilities and not found a counter-
0   0          1       example. The two propositions must be logically equivalent.
0   1          0   •   Note: Given such equivalence, if and only if or iff is also stated
1   0          0       as is a necessary and sufficient condition for.
1   1          1


                               *Proof for (PQ)(QP)  (P  Q)

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3.1. Propositional Equivalences

   Some important logical equivalences:
       Equivalences      Name
                                                        P T   PT

       PT  P                                          0 1    1
                         Identity laws
       PF  P
                                                        1 1    1



       PT  T                                          P P   PP
                         Domination laws
       PF  F
                                                        0 0    0
                                                        1 1    1

       PP  P
                         Idempotent laws
       PP  P
                         Double negation
       (P)  P         law
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3.1. Propositional Equivalences

   Some important logical equivalences:
       Equivalences                     Name
       PQ  QP
                                        Commutative laws
       PQ  QP
       (PQ)R  P(QR)
                                        Associative laws
       (PQ)R  P(QR)
       P(QR)  (PQ)(PR)
                                        Distributive laws
       P(QR)  (PQ)(PR)
       (PQ)  PQ
                                        De Morgan’s laws
       (PQ)  PQ
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3.1. Propositional Equivalences

   Other logical equivalences:
       Equivalences                     Name
       PQ  PQ                       Implication
       PP  T                         Tautology
       PP  F                         Contradiction
       (PQ)(QP)  (PQ)              Equivalence
       (PQ)(PQ)  P                Absurdity



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    3.1. Propositional Equivalences

   Other logical equivalences:
           Equivalences                     Name
           (PQ)  (QP)                  Contrapositive
           P(PQ)  P
                                            Absorption
           P (PQ)  P
           (P  Q)R  P(QR)              Exportation
   Equivalent expressions can always be substituted
    for each other in a more complex expression
      – useful for simplification.
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3.1. Propositional Equivalences
   Example:
       •   (P(PQ)) can be simplified by using the following
           series of logical equivalence:
               (P(PQ))
               P(PQ))         from the second De Morgan’s law
                P[(P)Q] from the first De Morgan’s law
                P(PQ)          from the double negation law
                (PP)(PQ] from the distributive law
                F(PQ)          since PPF
                PQ              from the identity law for F
                (PQ)             from the second De Morgan’s law



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 3.1. Propositional Equivalences


                        REMEMBER!

We can always use a truth table to show that the simplified
proposition is equivalent to the original proposition.



    But Complexity (2n)…



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3.2. Predicates and Quantifiers

   2+1=3: a proposition
   x+y=3: propositional functions or predicates
       •   A generalization of propositions
       •   Propositions which contain variables
       •   Predicates become propositions once every variable is
           bound - by
            • assigning it a value from the Universe of Discourse U
            or
            • quantifying it


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3.2.1. Predicates

   Example 1:
       •   Let U = Z, the integers = , -2, -1, 0 , 1, 2, 3, 
       •   P(x): x > 0, a predicate or propositional function.
       •   It has no truth value until the variable x is bound.
       •   Examples of propositions where x is assigned a value:
            • P(-3) is false, i.e. -3 > 0 is false.
            • P(0) is false.
            • P(3) is true.
       •   The collection of integers for which P(x) is true are the
           positive integers.


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3.2.1. Predicates

   Example 1 (continued):
       •   P(x): x > 0 is the predicate.
            • P(y)P(0) is not a proposition. The variable y has not
                been bound.
            •   However, P(3)P(0) is a proposition which is true.
   Example 2:
       •   Let R be the three-variable predicate R(x, y, z): x + y =
           z. Find the truth value of
            • R(2, -1, 5)
            • R(3, 4, 7)
            • R(x, 3, z)

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3.2.2. Quantifiers

   Specific value vs. Range
   What range of values in U for which the
    bounded propositions are true?
   Two possibilities:
       •   Universal: For all values in U
       •   Existential: For some values in U




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3.2.2. Quantifiers
   Universal
       •   P(x) is true for every x in the universe of discourse.
       •   Notation: universal quantifier x P(x)
            • For all x, P(x)
            or
            • For every x, P(x)
       •   The variable x is bound by the universal quantifier
           producing a proposition.
       •   Example:
            • U = { 1,2,3 }
                     x P(x)  P(1)P(2)P(3)


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3.2.2. Quantifiers
   Existential
       •   P(x) is true for some x in the universe of discourse.
       •   Notation: existential quantifier x P(x)
            • There is an x such that P(x)
            or
            • For some x, P(x)
            • For at least one x, P(x)
            • I can find an x such that P(x)
       •   Example:
            • U = { 1,2,3 }
                    x P(x)  P(1)P(2)P(3)



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3.2.2. Quantifiers
                               REMEMBER!
A predicate (propositional function) is not a proposition until all variables
have been bound either by quantification or assignment of a value!


   Predicate equivalences:
       • Equivalences involving the negation operator
          • x P(x)  x P(x)
          • x P(x)  x P(x)
       • Distributing a negation operator across a
        quantifier changes a universal to an
        existential, and vice versa.

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3.2.2. Quantifiers

   Multiple Quantifiers:
       •   Read from left to right . . .
       •   Example 1:
            • Let U = R, the real numbers,
                      P(x,y): xy = 0
                              xy P(x,y)
                              xy P(x,y)
                              xy P(x,y)
                              xy P(x,y)
            •   The only one that is false is the first one. Why?
            •   Suppose P(x,y) is the predicate x/y=1?

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3.2.2. Quantifiers

   Dangerous situations:
       • Commutativity of quantifiers
          xy P(x, y)  yx P(x, y)? YES!
          xy P(x, y)  yx P(x, y)? NO!
                      DIFFERENT MEANING!




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3.2.2. Quantifiers

   Example of non-commutativity of quantifiers:
       •   Let Q(x, y) denote “x + y = 0.”
       •   Are the truth values of the quantifications yx P(x, y)
           and xy P(x, y) the same?
   The answer is NO since:
       •   yx P(x, y) means “There is a real number y such
           that for all real numbers x, Q(x, y) is true.”
            • The statement is false. Why?
       •   x y P(x, y) means “For every real number x there is
           a real number y such that Q(x, y) is true.”
            • The statement is true.
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3.2.3. Converting from English

   (can be very difficult)
   Example 1:
       • Express the statement “If somebody is female
        and is a parent, then this person is someone’s
        mother” as a logical expression.
          • Let
                  F(x): x is female.
                  P(x): x is a parent
                  M(x, y): x is the mother of y.
          • The statement applies to all people.
                  x ((F(x)  (P(x))  y M(x, y))


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3.2.3. Converting from English

   Example 2:
       • Express the statement “Everyone has exactly
        one best friend” as a logical expression.
         • Let
                 B(x, y): y is the best friend of x.
         • The statement says “exactly one best friend”. This
           means that if y is the best friend of x, then all other
           people z other than y can not be the best friend of
           x.
                 xyz (B(x, y)  ((z  y)  B(x, z)))



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3.2.3. Converting from English

   Example 3:
       • Consider the following statements.
             “All lions are fierce.”
             “Some lions do not drink coffee.”
             “Some fierce creatures do not drink coffee.”
       •   The first two are called premises and the third
           is called the conclusion. The entire set is
           called an argument.



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3.2.3. Converting from English

   Example 3 (continued):
       • We can express these statements as follows.
         • Let    P(x): x is a lion.
                  Q(x): x is fierce.
                  R(x): x drinks coffee.
         • Then x (P(x)  Q(x)).
                  x (P(x)  R(x)).
                  x (Q(x)  R(x)).
       • Why can’t we write the second statement as
                 x (P(x)  R(x))?


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3.3. Further Readings

   Propositional Equivalences
       • Rosen: Section 1.2.
   Predicates and Quantifiers
       • Rosen: Section 1.3.




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