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					Propositional Logic

  CS 1050
  (Rosen Section 1.1, 1.2)
              Proposition
A proposition is a statement that is either
  true or false, but not both.
• Atlanta was the site of the 1996
  Summer Olympic games.
• 1+1 = 2
• 3+1 = 5
• What will my CS1050 grade be?
      Definition 1. Negation of p
   Let p be a proposition.
   The statement “It is
                               Table 1.
   not the case that p” is     The Truth Table for the
   also a proposition,         Negation of a Proposition

   called the “negation of
                                  p                 ¬p
   p” or ¬p (read “not p”)
p = The sky is blue.              T                 F
                                  F                 T
p = It is not the case that
the sky is blue.
p = The sky is not blue.
 Definition 2. Conjunction of p
             and q
Let p and q be           Table 2. The Truth Table for
propositions. The        the Conjunction of two
                         propositions
proposition “p and q,”
denoted by pq is true   p        q        pq
when both p and q are
true and is false         T      T           T
otherwise. This is        T      F            F
called the conjunction    F      T            F
                          F      F            F
of p and q.
    Definition 3. Disjunction of p
                and q
                               Let p and q be
Table 3. The Truth Table for
the Disjunction of two         propositions. The
propositions
                               proposition “p or q,”
                               denoted by pq, is the
p        q        pq
                               proposition that is false
 T      T           T          when p and q are both
 T      F           T          false and true otherwise.
 F      T           T
 F      F            F
 Definition 4. Exclusive or of p
              and q
                               Let p and q be
Table 4. The Truth Table for
the Exclusive OR of two        propositions. The
propositions
                               exclusive or of p and q,
                               denoted by pq, is the
p        q        pq
                               proposition that is true
 T      T           F          when exactly one of p
 T      F           T          and q is true and is
 F      T           T          false otherwise.
 F      F            F
   Definition 5. Implication pq
Let p and q be propositions.
                                 Table 5. The Truth Table for
The implication pq is the       the Implication of pq.
proposition that is false when
p is true and q is false, and
                                 p        q        pq
true otherwise. In this
implication p is called the       T      T           T
hypothesis (or antecedent or      T      F           F
premise) and q is called the      F      T           T
conclusion (or                    F      F           T
consequence).
                 Implications
•   If p, then q            • Not the same as the
•   p implies q               if-then construct
•   if p,q                    used in
                              programming
•   p only if q               languages such as
•   p is sufficient for q     If p then S
•   q if p
•   q whenever p
•   q is necessary for p
                   Implications
How can both p and q be false, and pq be true?
•Think of p as a “contract” and q as its “obligation” that is
only carried out if the contract is valid.
•Example: “If you make more than $25,000, then you must
file a tax return.” This says nothing about someone who
makes less than $25,000. So the implication is true no
matter what someone making less than $25,000 does.
•Another example:
         p: Bill Gates is poor.
         q: Pigs can fly.
pq is always true because Bill Gates is not poor. Another
way of saying the implication is
“Pigs can fly whenever Bill Gates is poor” which is true
since neither p nor q is true.
       Related Implications
Converse of      Contrapositive
 pq              of p  q
 is               is the proposition
 qp              q  p
     Definition 6. Biconditional
                               Let p and q be
Table 6. The Truth Table for   propositions. The
the biconditional pq.
                               biconditional pq is the
                               proposition that is true
p        q        pq
                               when p and q have the
 T      T           T          same truth values and is
 T      F           F          false otherwise. “p if and
 F      T           F          only if q, p is necessary
 F      F           T          and sufficient for q”
                      Practice
            p: You learn the simple things well.
            q: The difficult things become easy.
• You do not learn the        • The difficult things
  simple things well. p        become easy but you
• If you learn the simple       did not learn the simple
  things well then the          things well. q  p
  difficult things become     • You learn the simple
  easy.       pq               things well but the
• If you do not learn the       difficult things did not
  simple things well, then      become easy.
  the difficult things will            p  q
  not become easy.
          p  q
         Truth Table Puzzle
Steve would like to determine the relative
  salaries of three coworkers using two facts
  (all salaries are distinct):
• If Fred is not the highest paid of the three,
  then Janice is.
• If Janice is not the lowest paid, then Maggie
  is paid the most.
Who is paid the most and who is paid the least?
p : Janice is paid the most.     •If Fred is not the highest paid
q: Maggie is paid the most.      of the three, then Janice is.
                                 •If Janice is not the lowest paid,
r: Fred is paid the most.        then Maggie is paid the most.
s: Janice is paid the least.

p     q      r     s      rp    s q (rp) (sq)
T     F      F     F      T       F         F
F     T      F     T      F       T         F
F     F      T     T      T       T         T
F     T      F     F      F       T         F
F     F      T     F      T       F         F




Fred, Maggie, Janice
p : Janice is paid the most.     •If Fred is not the highest paid
q: Maggie is paid the most.      of the three, then Janice is.
                                 •If Janice is the lowest paid,
r: Fred is paid the most.        then Maggie is paid the most.
s: Janice is paid the least.

p     q      r     s      rp    s q       (rp) (sq)
T     F      F     F      T        T            T
F     T      F     T      F        T            F
F     F      T     T      T        F            F
F     T      F     F      F        T            F
F     F      T     F      T        T            T



Fred, Janice, Maggie or Janice, Maggie, Fred
or Janice, Fred, Maggie
                 Bit Operations
A computer bit has two possible values: 0 (false) and 1
(true). A variable is called a Boolean variable is its value is
either true or false.

Bit operations correspond to the logical connectives:
 OR
 AND
 XOR

Information can be represented by bit strings, which are
sequences of zeros and ones, and manipulated by
operations on the bit strings.
    Truth tables for the bit
operations OR, AND, and XOR
     0   1
                             0   1

                          0   0 1
  0   0 1

  1   1 1                 1   1 0
                 0   1

              0   0 0

              1   0 1
         Logical Equivalence
• An important technique in proofs is to
  replace a statement with another
  statement that is “logically equivalent.”
• Tautology: compound proposition that is
  always true regardless of the truth values
  of the propositions in it.
• Contradiction: Compound proposition
  that is always false regardless of the truth
  values of the propositions in it.
       Logically Equivalent
• Compound propositions P and Q are
  logically equivalent if PQ is a
  tautology. In other words, P and Q
  have the same truth values for all
  combinations of truth values of simple
  propositions.
• This is denoted: PQ (or by P  Q)
      Example: DeMorgans
• Prove that (pq)  (p  q)
p q   (pq)   (pq) p q   (p  q)
TT     T        F    F   F      F
TF     T        F    F   T      F
FT     T        F    T   F      F
FF     F        T    T   T      T
Illustration of De Morgan’s Law
       (pq)




          p
                   q
Illustration of De Morgan’s Law
       p




            p
Illustration of De Morgan’s Law
       q




                   q
Illustration of De Morgan’s Law
       p  q




          p
                   q
             Example: Distribution
Prove that: p  (q  r)  (p  q)  (p  r)

    p   q    r   qr p(qr) pq pr      (pq)(pr)
    T   T   T    T    T      T   T           T
    T   T   F     F    T      T   T           T
    T   F   T     F    T      T   T          T
    T   F   F     F    T      T   T           T
    F   T   T     T   T       T   T          T
    F   T   F     F    F      T    F          F
    F   F   T     F    F      F    T          F
    F   F   F     F    F      F    F          F
 Prove: pq(pq)  (qp)
pq    pq     pq qp              (pq)(qp)
TT    T       T   T                  T
TF    F       F   T                  F
FT    F       T   F                  F
FF    T       T   T                  T

We call this biconditional equivalence.
  List of Logical Equivalences
pT  p;    pF  p            Identity Laws

pT  T;    pF  F            Domination Laws

pp  p;    pp  p            Idempotent Laws

(p)  p                      Double Negation Law

pq  qp; pq  qp           Commutative Laws

(pq) r  p (qr); (pq)  r  p  (qr)
                                  Associative Laws
          List of Equivalences
p(qr)  (pq)(pr)   Distribution Laws
p(qr)  (pq)(pr)

(pq)(p  q)        De Morgan’s Laws
(pq)(p  q)
                        Miscellaneous
p  p  T              Or Tautology
p  p  F              And Contradiction
(pq)  (p  q)        Implication Equivalence

pq(pq)  (qp)       Biconditional Equivalence

				
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posted:8/28/2011
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