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Discrete Mathematics
5th edition, 2001
Chapter 1
Logic and proofs
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Logic
Logic = the study of correct reasoning
Use of logic In mathematics:
to prove theorems
In computer science:
to prove that programs do what they are
supposed to do
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Section 1.1 Propositions
A propositionis a statement or sentence
that can be determined to be either true orfalse.
Examples:
John is a programmer" is a proposition I wish I were wise is not a proposition
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Connectives
If p and q are propositions, new compound
propositions can be formed by using
connectives
Most common connectives: Conjunction AND. Symbol ^
Inclusive disjunction OR Symbol v
Exclusive disjunction OR Symbol v
Negation Symbol ~
Implication Symbol
Double implication Symbol
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Truth table of conjunction
The truth values of compound propositionscan be described by truth tables.
Truth table ofconjunction
p ^ q is true only when both p and q are true.
p q p ^ q
T T T
T F F
F T F
F F F
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Example
Let p = Tigers are wild animals
Let q = Chicago is the capital of Illinois p ^ q = "Tigers are wild animals and
Chicago is the capital of Illinois"
p ^ q is false. Why?
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Truth table of disjunction
The truth table of (inclusive) disjunctionis
p q is false only when both p and q are false Example: p = "John is a programmer", q = "Mary is a lawyer"
p v q = "John is a programmer or Mary is a lawyer"
p q p v q
T T T
T F T
F T T
F F F
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Exclusive disjunction
Either p or q (but not both), in symbols p q
p q is true only when p is true and q is false,or p is false and q is true. Example: p = "John is programmer, q = "Mary is a lawyer"
p v q = "Either John is a programmer or Mary is a lawyer"
p q p v q
T T F
T F T
F T T
F F F
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Negation
Negation of p: in symbols ~p
~p is false when p is true, ~p is true when p isfalse Example: p = "John is a programmer"
~p = "It is not true that John is a programmer"
p ~p
T F
F T
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More compound statements
Let p, q, r be simple statements
We can form other compound statements,
such as (pq)^r
p(q^r)
(~p)
(~q) (pq)^(~r)
and many others
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Example: truth table of (pq)^r
p q r (p 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 FF F T F
F F F F
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1.2 Conditional propositions
and logical equivalence
A conditionalproposition is of the form
If p then q In symbols: p q
Example: p = " John is a programmer"
q = " Mary is a lawyer "
p q = If John is a programmer then Mary isa lawyer"
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Truth table of p q
p q is true when both p and q are true
or when p is false
p q p q
T T T
T F F
F T T
F F T
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Hypothesis and conclusion
In a conditional proposition p
q,p is called the antecedentorhypothesis
q is called the consequentorconclusion
If "p then q" is considered logically the
same as "p only if q"
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Necessary and sufficient
A necessarycondition is expressed by the
conclusion.
A sufficientcondition is expressed by the
hypothesis.
Example:
IfJohn is a programmerthenMary is a lawyer"
Necessary condition: Mary is a lawyer Sufficient condition: John is a programmer
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Logical equivalence
Two propositions are said to be logically
equivalentif their truth tables are identical.
Example: ~p q is logically equivalentto p q
p q ~p q p q
T T T T
T F F F
F T T TF F T T
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Converse
The converseof p q is q p
These two propositions
are not logically equivalent
p q p q q p
T T T T
T F F T
F T T F
F F T T
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Contrapositive
The contrapositiveof the proposition p q is
~q ~p.
They are logically equivalent.
p q p q ~q ~p
T T T T
T F F F
F T T T
F F T T
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Double implication
The double implicationp if and only if q isdefined in symbols as p q
p q is logically equivalent to (p q)^(q p)
p q p q (p q) ^ (q p)
T T T T
T F F F
F T F F
F F T T
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Tautology
A proposition is a tautologyif its truth table
contains only true values for every case
Example: p p v q
p q p p v q
T T T
T F T
F T T
F F T
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Contradiction
A proposition is a tautologyif its truth table
contains only false values for every case
Example: p ^ ~p
p p ^ (~p)
T F
F F
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De Morgans laws for logic
The following pairs of propositions are
logically equivalent:
~ (p q) and (~p)^(~q)
~ (p ^ q) and (~p) (~q)
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1.3 Quantifiers
A propositional functionP(x) is a statement
involving a variable x
For example: P(x): 2x is an even integer
x is an element of a set D
For example, x is an element of the set of integers
D is called the domainof P(x)
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Domain of a propositional function
In the propositional function
P(x): 2x is an even integer,the domain D of P(x) must be defined, for
instance D = {integers}.
D is the set where the x's come from.
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For everyand for some
Most statements in mathematics and
computer science use terms such as for
everyand for some.
For example:
For everytriangle T, the sum of the angles of T
is 180 degrees.
For everyinteger n, n is less than p, for someprime number p.
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Universal quantifier
One can write P(x) for everyx in a domain D
In symbols: x P(x)
is called the universal quantifier
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Truth of as propositional function
The statement x P(x) is
True if P(x) is true for every x D
False if P(x) is not true for some x
D Example: Let P(n) be the propositional
function n2 + 2n is an odd integer
n D = {all integers}
P(n) is true only when n is an odd integer,
false if n is an even integer.
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Existential quantifier
For somex D, P(x) is true ifthere exists
an element x in the domain D for which P(x) is
true. In symbols: x, P(x)
The symbol is called the existential
quantifier.
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Counterexample
The universal statement x P(x) is false ifx D such that P(x) is false.
The value x that makes P(x) false is called acounterexampleto the statement x P(x). Example: P(x) = "every x is a prime number", for
every integer x.
But if x = 4 (an integer) this x is not a primernumber. Then 4 is a counterexample to P(x)being true.
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Generalized De Morgans
laws for Logic 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 equivalentto "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) istrue" is equivalent to "For all x, P(x) is not true"
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Summary of propositional logic
In order to prove the
universally quantified
statement x P(x) is
true It is not enough to
show P(x) true for
some x D
You must show P(x) istrue for every x D
In order to prove the
universally quantified
statement x P(x) is
false It is enough to exhibit
some x D for which
P(x) is false
This x is called thecounterexample to
the statement x P(x)
is true
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1.4 Proofs
A mathematical systemconsists of
Undefined terms Definitions
Axioms
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Undefined terms
Undefined termsare the basic building blocks of
a mathematical system. These are words that
are accepted as starting concepts of amathematical system.
Example: in Euclidean geometry we have undefined
terms such as
Point
Line
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Definitions
A definitionis a proposition constructed from
undefined terms and previously accepted
concepts in order to create a new concept.
Example. In Euclidean geometry the followingare definitions:
Two triangles are congruentif their vertices can
be paired so that the corresponding sides are
equal and so are the corresponding angles. Two angles are supplementaryif the sum of their
measures is 180 degrees.
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Axioms
An axiomis a proposition accepted as true
without proof within the mathematical system.
There are many examples of axioms in
mathematics:
Example: In Euclidean geometry the following are
axioms
Given two distinct points, there is exactly one line that
contains them.
Given a line and a point not on the line, there is exactly one
line through the point which is parallel to the line.
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Theorems
A theoremis a proposition of the form p q
which must be shown to be true by asequence of logical steps that assume that p
is true, and use definitions, axioms and
previously proven theorems.
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Lemmas and corollaries
A lemmais a small theorem which is
used to prove a bigger theorem.
A corollaryis a theorem that can be
proven to be a logical consequence of
another theorem.
Example from Euclidean geometry: "If the
three sides of a triangle have equal length,
then its angles also have equal measure."
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Types of proof
A proofis a logical argument that consists of a
series of steps using propositions in such a
way that the truth of the theorem isestablished.
Directproof: p q
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.
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Indirect proof
The method of proofby contradictionof atheorem p q consists of the followingsteps:
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) isalways false. There is a contradiction!
5. So, q cannot be false and therefore it is true.
OR: show that the contrapositive(~q)(~p)is true. Since (~q) (~p) is logically equivalent to p q, then the
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Valid arguments
Deductive reasoning: the process of reaching a
conclusion q from a sequence of propositions p1,
p2, , pn. The propositions p1, p2, , pn are called
premisesorhypothesis.
The proposition q that is logically obtainedthrough the process is called the conclusion.
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Rules of inference (1)
1. Law ofdetachmentor
modus ponens p q
p
Therefore, q
2. Modus tollens
p q ~q
Therefore, ~p
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Rules of inference (2)
3. Rule ofAddition
p
Therefore, p q
4. Rule ofsimplification
p ^ q
Therefore, p
5. Rule ofconjunction
p
q
Therefore, p ^ q
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Rules of inference (3)
6. Rule ofhypothetical syllogism
p q
q r
Therefore, p r
7. Rule ofdisjunctive syllogism
p q
~p
Therefore, q
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Rules of inference for
quantified statements
1. Universal instantiation
xD, P(x) d D
Therefore P(d)
2. Universal generalization
P(d) for any d D
Therefore x, P(x)
3. Existential instantiation
x D, P(x) Therefore P(d) for some
d D
4. Existential generalization
P(d) for some d D Therefore x, P(x)
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1.5 Resolution proofs
Due to J. A. Robinson (1965)
A clauseis a compound statement with terms separated
by or, and each term is a single variable or the
negation of a single variable
Example: p q (~r) is a clause
(p ^ q) r (~s) is not a clause
Hypothesis and conclusion are written as clauses
Only one rule:
p q
~p r
Therefore, q r
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1.6 Mathematical induction
Useful for proving statements of the form
n A S(n)
where N is the set of positive integers or naturalnumbers,
A is an infinite subset of N
S(n) is a propositional function
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Mathematical Induction:
strong form Suppose we want to show that for each positive
integer n the statement S(n) is either true or
false. 1. Verify that S(1) is true.
2. Let n be an arbitrary positive integer. Let i be a
positive integer such that i < n.
3. Show that S(i) true implies that S(i+1) is true, i.e.
show S(i) S(i+1).
4. Then conclude that S(n) is true for all positive
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Mathematical induction:
terminology
Basis step: Verify that S(1) is true.
Inductive step: Assume S(i) is true.
Prove S(i) S(i+1).
Conclusion: Therefore S(n) is true for all
positive integers n.
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