R. Johnsonbaugh_ Discrete Mathematics 5th edition_ 2001 - metalab by yurtgc548

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									       R. Johnsonbaugh,
Discrete Mathematics
                      5th edition, 2001


      Chapter 1
   Logic and proofs
                    Logic

p Logic = the study of correct reasoning
p Use of logic
    n   In mathematics:
         p   to prove theorems
    n   In computer science:
         p   to prove that programs do what they are
             supposed to do
Section 1.1 Propositions


p A proposition is a statement or sentence
  that can be determined to be either true or
  false.
p Examples:
    n   “John is a programmer" is a proposition
    n   “I wish I were wise” is not a proposition
             Connectives
If p and q are propositions, new compound
   propositions can be formed by using
   connectives
p Most common connectives:
  n   Conjunction AND.           Symbol ^
  n   Inclusive disjunction OR   Symbol v
  n   Exclusive disjunction OR   Symbol v
  n   Negation                   Symbol ~
  n   Implication                Symbol ®
  n   Double implication         Symbol «
    Truth table of conjunction
p The truth values of compound propositions
  can be described by truth tables.
p Truth table of conjunction


                p    q      p^q
                T    T       T
                T    F       F
                F    T       F
                F    F       F

p   p ^ q is true only when both p and q are true.
            Example

p Let p = “Tigers are wild animals”
p Let q = “Chicago is the capital of Illinois”
p p ^ q = "Tigers are wild animals and
  Chicago is the capital of Illinois"
p p ^ q is false. Why?
     Truth table of disjunction
q   The truth table of (inclusive) disjunction is
                                 p     q      pvq
                                 T     T       T
                                 T     F         T
                                 F     T         T
                                 F     F         F

q   p Ú q is false only when both p and q are false
    q   Example: p = "John is a programmer", q = "Mary is a lawyer"
    q   p v q = "John is a programmer or Mary is a lawyer"
          Exclusive disjunction
p   “Either p or q” (but not both), in symbols p Ú q
                                        p     q      pvq
                                        T     T       F
                                        T     F        T
                                        F     T        T
                                        F     F        F

q   p Ú q is true only when p is true and q is false,
    or p is false and q is true.
    q   Example: p = "John is programmer, q = "Mary is a lawyer"
    q   p v q = "Either John is a programmer or Mary is a lawyer"
                       Negation
p   Negation of p: in symbols ~p
                                   p      ~p
                                   T       F

                                   F       T


p   ~p is false when p is true, ~p is true when p is
    false
    n   Example: p = "John is a programmer"
    n   ~p = "It is not true that John is a programmer"
More compound statements

p Let p, q, r be simple statements
p We can form other compound statements,
  such as
    n   (pÚq)^r
    n   pÚ(q^r)
    n   (~p)Ú(~q)
    n   (pÚq)^(~r)
    n   and many others…
Example: truth table of (pÚq)^r
     p   q    r   (p Ú q) ^ r
     T   T    T       T
     T   T    F       F
     T   F    T       T
     T   F    F       F
     F   T    T       T
     F   T    F       F
     F   F    T       F
     F   F    F       F
1.2 Conditional propositions
   and logical equivalence
p A conditional proposition is of the form
                “If p then q”
p In symbols: p ® q
p Example:
    n   p = " John is a programmer"
    n   q = " Mary is a lawyer "
    n   p ® q = “If John is a programmer then Mary is
        a lawyer"
     Truth table of p ® q

               p    q    p®q
               T    T      T
               T    F      F

               F    T      T
               F    F      T

q   p ® q is true when both p and q are true
               or when p is false
Hypothesis and conclusion


p In a conditional proposition p ® q,
     p is called the antecedent or hypothesis
     q is called the consequent or conclusion
q If "p then q" is considered logically the
  same as "p only if q"
    Necessary and sufficient
p A necessary condition is expressed by the
  conclusion.
p A sufficient condition is expressed by the
  hypothesis.
    n   Example:
         If John is a programmer then Mary is a lawyer"
    n   Necessary condition: “Mary is a lawyer”
    n   Sufficient condition: “John is a programmer”
           Logical equivalence
q   Two propositions are said to be logically
    equivalent if their truth tables are identical.

           p          q       ~p Ú q     p®q

           T          T          T         T
           T          F          F         F
           F          T          T         T
           F          F          T         T

q   Example: ~p Ú q is logically equivalent to p ® q
                Converse
p   The converse of p ® q is q ® p

            p      q      p®q      q®p
            T      T       T        T
            T      F        F           T
            F      T        T           F
            F      F        T           T


           These two propositions
         are not logically equivalent
                Contrapositive
p   The contrapositive of the proposition p ® q is
    ~q ® ~p.

         p      q       p®q         ~q ® ~p
         T      T         T            T
         T      F         F            F
         F      T         T            T
         F      F         T            T


             They are logically equivalent.
             Double implication
p   The double implication “p if and only if q” is
    defined in symbols as p « q

             p      q     p«q      (p ® q) ^ (q ® p)
             T      T        T             T
             T      F        F             F
             F      T        F             F
             F      F        T             T

    p « q is logically equivalent to (p ® q)^(q ® p)
                      Tautology
p   A proposition is a tautology if its truth table
    contains only true values for every case
    n   Example: p ® p v q

                  p          q   p®pvq
                  T          T       T
                  T          F       T
                  F          T       T
                  F          F       T
                   Contradiction
p   A proposition is a tautology if its truth table
    contains only false values for every case
    n   Example: p ^ ~p

                              p     p ^ (~p)
                              T        F
                              F        F
De Morgan’s laws for logic

p   The following pairs of propositions are
    logically equivalent:

    n ~ (p Ú q) and (~p)^(~q)
    n ~ (p ^ q) and (~p) Ú (~q)
               1.3 Quantifiers

p A propositional function P(x) is a statement
  involving a variable x
p For example:
    n   P(x): 2x is an even integer
p   x is an element of a set D
    n   For example, x is an element of the set of integers
p   D is called the domain of P(x)
Domain of a propositional function

  p In the propositional function
          P(x): “2x is an even integer”,
  the domain D of P(x) must be defined, for
  instance D = {integers}.
  p D is the set where the x's come from.
    For every and for some
p Most statements in mathematics and
  computer science use terms such as for
  every and for some.
p For example:
    n   For every triangle T, the sum of the angles of T
        is 180 degrees.
    n   For every integer n, n is less than p, for some
        prime number p.
          Universal quantifier

p   One can write P(x) for every x in a domain D
    n   In symbols: "x P(x)
p   " is called the universal quantifier
Truth of as propositional function

p   The statement "x P(x) is
    n   True if P(x) is true for every x Î D
    n   False if P(x) is not true for some x Î D
p Example: Let P(n) be the propositional
  function n2 + 2n is an odd integer
  "n Î D = {all integers}
p P(n) is true only when n is an odd integer,
  false if n is an even integer.
       Existential quantifier

p  For some x Î D, P(x) is true if there exists
an element x in the domain D for which P(x) is
true. In symbols: $x, P(x)

p   The symbol $ is called the existential
    quantifier.
            Counterexample
p   The universal statement "x P(x) is false if
    $x Î D such that P(x) is false.

p   The value x that makes P(x) false is called a
    counterexample to the statement "x P(x).
    n   Example: P(x) = "every x is a prime number", for
        every integer x.
    n   But if x = 4 (an integer) this x is not a primer
        number. Then 4 is a counterexample to P(x)
        being true.
    Generalized De Morgan’s
         laws for Logic
p    If P(x) is a propositional function, then each
    pair of propositions in a) and b) below have
    the same truth values:
          a) ~("x P(x)) and $x: ~P(x)
    "It is not true that for every x, P(x) holds" is equivalent
    to "There exists an x for which P(x) is not true"
          b) ~($x P(x)) and "x: ~P(x)
    "It is not true that there exists an x for which P(x) is
    true" is equivalent to "For all x, P(x) is not true"
Summary of propositional logic
p   In order to prove the       p   In order to prove the
    universally quantified          universally quantified
    statement "x P(x) is            statement "x P(x) is
    true                            false
    n   It is not enough to         n   It is enough to exhibit
        show P(x) true for              some x Î D for which
        some x Î D                      P(x) is false
    n   You must show P(x) is       n   This x is called the
        true for every x Î D            counterexample to
                                        the statement "x P(x)
                                        is true
             1.4 Proofs

p   A mathematical system consists of
    n Undefined terms
    n Definitions
    n Axioms
                Undefined terms

p   Undefined terms are the basic building blocks of
    a mathematical system. These are words that
    are accepted as starting concepts of a
    mathematical system.
    n   Example: in Euclidean geometry we have undefined
        terms such as
          p Point

          p Line
                   Definitions
p   A definition is a proposition constructed from
    undefined terms and previously accepted
    concepts in order to create a new concept.
    n   Example. In Euclidean geometry the following
        are definitions:
    n   Two triangles are congruent if their vertices can
        be paired so that the corresponding sides are
        equal and so are the corresponding angles.
    n   Two angles are supplementary if the sum of their
        measures is 180 degrees.
                               Axioms
p An axiom is a proposition accepted as true
  without proof within the mathematical system.
p There are many examples of axioms in
  mathematics:
    n   Example: In Euclidean geometry the following are
        axioms
         p   Given two distinct points, there is exactly one line that
             contains them.
         p   Given a line and a point not on the line, there is exactly one
             line through the point which is parallel to the line.
               Theorems

p   A theorem is a proposition of the form p ® q
    which must be shown to be true by a
    sequence of logical steps that assume that p
    is true, and use definitions, axioms and
    previously proven theorems.
Lemmas and corollaries

p   A lemma is a small theorem which is
    used to prove a bigger theorem.

p   A corollary is a theorem that can be
    proven to be a logical consequence of
    another theorem.
    n   Example from Euclidean geometry: "If the
        three sides of a triangle have equal length,
        then its angles also have equal measure."
                  Types of proof

p A proof is a logical argument that consists of a
  series of steps using propositions in such a
  way that the truth of the theorem is
  established.
p Direct proof: p ® q
    n   A direct method of attack that assumes the truth of
        proposition p, axioms and proven theorems so that
        the truth of proposition q is obtained.
                  Indirect proof
q Themethod of proof by contradiction of a
 theorem p ® q consists of the following
 steps:
  1. Assume p is true and q is false
  2. Show that ~p is also true.
  3. Then we have that p ^ (~p) is true.
  4. But this is impossible, since the statement p ^ (~p) is
    always false. There is a contradiction!
  5. So, q cannot be false and therefore it is true.
q OR:  show that the contrapositive (~q)®(~p)
 is true.
  q   Since (~q) ® (~p) is logically equivalent to p ® q, then the
      theorem is proved.
            Valid arguments

p Deductive reasoning: the process of reaching a
  conclusion q from a sequence of propositions p1,
  p2, …, pn.
p The propositions p1, p2, …, pn are called
  premises or hypothesis.
p The proposition q that is logically obtained
  through the process is called the conclusion.
          Rules of inference (1)

1. Law of detachment or   2. Modus tollens
  modus ponens              n   p®q
  n   p®q                   n   ~q
  n   p                     n   Therefore, ~p
  n   Therefore, q
           Rules of inference (2)
3. Rule of Addition
   n   p                    5. Rule of conjunction
   n   Therefore, p Ú q       n   p
                              n   q
4. Rule of simplification     n   Therefore, p ^ q
   n   p^q
   n   Therefore, p
  Rules of inference (3)
6. Rule of hypothetical syllogism
   np®q
   nq®r
   n Therefore, p ® r


7. Rule of disjunctive syllogism
   npÚq
   n ~p
   n Therefore, q
           Rules of inference for
           quantified statements
1. Universal instantiation  3. Existential instantiation
    n " xÎD, P(x)               n $ x Î D, P(x)
    n dÎD                       n Therefore P(d) for some
    n Therefore P(d)              d ÎD
2. Universal generalization 4. Existential generalization
    n P(d) for any d Î D        n P(d) for some d ÎD

    n Therefore "x, P(x)        n Therefore $ x, P(x)
            1.5 Resolution proofs
p   Due to J. A. Robinson (1965)
p   A clause is a compound statement with terms separated
    by “or”, and each term is a single variable or the
    negation of a single variable
    n   Example: p Ú q Ú (~r) is a clause
                (p ^ q) Ú r Ú (~s) is not a clause
p   Hypothesis and conclusion are written as clauses
p   Only one rule:
    n   pÚq
    n   ~p Ú r
    n   Therefore, q Ú r
     1.6 Mathematical induction

p   Useful for proving statements of the form
    " n Î A S(n)
    where N is the set of positive integers or natural
    numbers,
    A is an infinite subset of N
    S(n) is a propositional function
          Mathematical Induction:
               strong form
q   Suppose we want to show that for each positive
    integer n the statement S(n) is either true or
    false.
    n   1. Verify that S(1) is true.
    n   2. Let n be an arbitrary positive integer. Let i be a
        positive integer such that i < n.
    n   3. Show that S(i) true implies that S(i+1) is true, i.e.
        show S(i) ® S(i+1).
    n   4. Then conclude that S(n) is true for all positive
        integers n.
        Mathematical induction:
             terminology

p Basis step:       Verify that S(1) is true.
p Inductive step:   Assume S(i) is true.
                    Prove S(i) ® S(i+1).
p   Conclusion:     Therefore S(n) is true for all
                    positive integers n.

								
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