# 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
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)
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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