Ch01 logic 7238248327433333722u393-1.ppt

Spring 2003 CMSC 203 - Discrete Structures 1
Let’s get started with...
Let’s get started with...
Logic
Logic!
!

Logic
Logic
• Crucial for mathematical reasoning
Crucial for mathematical reasoning
• Important for program design
Important for program design
• Used for designing electronic circuitry
Used for designing electronic circuitry
• (Propositional )Logic is a system based on
(Propositional )Logic is a system based on
propositions
propositions.
.
• A proposition is a (declarative) statement that is
A proposition is a (declarative) statement that is
either
either true
true or
or false
false (not both).
(not both).
• We say that the
We say that the truth value
truth value of a proposition is
of a proposition is
either true (
either true (T
T) or false (
) or false (F
F).
).
• Corresponds to
Corresponds to 1
1 and
and 0
0 in digital circuits
in digital circuits

The Statement/Proposition Game
“
“Elephants are bigger than mice.”
Elephants are bigger than mice.”
Is this a statement?
Is this a statement? yes
yes
Is this a proposition?
Is this a proposition? yes
yes
What is the truth value
of the proposition?
of the proposition? true
true

“
“520 < 111”
520 < 111”
yes
yes
of the proposition?
of the proposition? false
false

“
“y > 5”
y > 5”
yes
Is this a proposition? no
no
Its truth value depends on the value of y,
Its truth value depends on the value of y,
but this value is not specified.
but this value is not specified.
We call this type of statement a
We call this type of statement a
propositional function
propositional function or
or open sentence
open sentence.
.

“
“Today is January 27 and 99 < 5.”
Today is January 27 and 99 < 5.”
yes
yes
of the proposition?
of the proposition? false
false

“
“Please do not fall asleep.”
Please do not fall asleep.”
Is this a statement? no
no
Is this a proposition? no
no
Only statements can be propositions.
Only statements can be propositions.
It’s a request.
It’s a request.

“
“If the moon is made of cheese,
If the moon is made of cheese,
then I will be rich.”
then I will be rich.”
yes
yes
of the proposition?
of the proposition? probably true
probably true

“
“x < y if and only if y > x.”
x < y if and only if y > x.”
yes
yes
of the proposition?
of the proposition? true
true
…
… because its truth value
because its truth value
does not depend on
does not depend on
specific values of x and y.
specific values of x and y.

Combining Propositions
Combining Propositions
As we have seen in the previous examples,
As we have seen in the previous examples,
one or more propositions can be combined
one or more propositions can be combined
to form a single
to form a single compound proposition
compound proposition.
.
We formalize this by denoting propositions
We formalize this by denoting propositions
with letters such as
with letters such as p, q, r, s,
p, q, r, s, and
and
introducing several
introducing several logical operators or
logical operators or
logical connectives
logical connectives.
.

Logical Operators (Connectives)
Logical Operators (Connectives)
We will examine the following logical operators:
We will examine the following logical operators:
• Negation
Negation (NOT,
(NOT, 
)
)
• Conjunction
Conjunction (AND,
(AND, 
)
)
• Disjunction
Disjunction (OR,
(OR, 
)
)
• Exclusive-or
Exclusive-or (XOR,
(XOR, 
 )
)
• Implication
Implication (if – then,
(if – then, 
 )
)
• Biconditional
Biconditional (if and only if,
(if and only if, 
 )
)
Truth tables can be used to show how these
Truth tables can be used to show how these
operators can combine propositions to compound
operators can combine propositions to compound
propositions.
propositions.

Negation (NOT)
Negation (NOT)
Unary Operator, Symbol:
Unary Operator, Symbol: 

P
P 
 P
P
true (T)
true (T) false (F)
false (F)
false (F)
false (F) true (T)
true (T)

Conjunction (AND)
Conjunction (AND)
Binary Operator, Symbol:
Binary Operator, Symbol: 

P
P Q
Q P
P
 Q
Q
T
T T
T T
T
T
T F
F F
F
F
F T
T F
F
F
F F
F F
F

Disjunction (OR)
Disjunction (OR)
Binary Operator, Symbol: 

P
P Q
Q P
P 
 Q
Q
T
T T
T T
T
T
T F
F T
T
F
F T
T T
T
F
F F
F F
F

Exclusive Or (XOR)
Exclusive Or (XOR)
Binary Operator, Symbol: 

P
P Q
Q P
P
Q
Q
T
T T
T F
F
T
T F
F T
T
F
F T
T T
T
F
F F
F F
F

Implication (if - then)
Implication (if - then)
Binary Operator, Symbol: 

P
P Q
Q P
P
Q
Q
T
T T
T T
T
T
T F
F F
F
F
F T
T T
T
F
F F
F T
T

Biconditional (if and only if)
Biconditional (if and only if)
Binary Operator, Symbol: 

P
P Q
Q P
P
Q
Q
T
T T
T T
T
T
T F
F F
F
F
F T
T F
F
F
F F
F T
T

Statements and Operators
Statements and Operators
Statements and operators can be combined in any
way to form new statements.
P
P Q
Q 
P
P 
Q
Q (
(
P)
P)
(
(
Q)
Q)
T
T T
T F
F F
F F
F
T
T F
F F
F T
T T
T
F
F T
T T
T F
F T
T
F
F F
F T
T T
T T
T

Statements and Operations
Statements and Operations
P
P Q
Q P
P
Q
Q 
(P
(P
Q)
Q) (
(
P)
P)
(
(
Q)
Q)
T
T T
T T
T F
F F
F
T
T F
F F
F T
T T
T
F
F T
T F
F T
T T
T
F
F F
F F
F T
T T
T

Exercises
Exercises
• To take discrete mathematics, you must have
To take discrete mathematics, you must have
taken calculus or a course in computer science.
• When you buy a new car from Acme Motor
When you buy a new car from Acme Motor
Company, you get $2000 back in cash or a 2%
car loan.
car loan.
• School is closed if more than 2 feet of snow
School is closed if more than 2 feet of snow
falls or if the wind chill is below -100.

Exercises
Exercises
– P: take discrete mathematics
P: take discrete mathematics
– Q: take calculus
Q: take calculus
– R: take a course in computer science
R: take a course in computer science
• P
P 
 Q
Q 
 R
R
• Problem with proposition R
Problem with proposition R
– What if I want to represent “take CMSC201”?
What if I want to represent “take CMSC201”?
• To take discrete mathematics, you must have
To take discrete mathematics, you must have

Exercises
Exercises
– P: buy a car from Acme Motor Company
P: buy a car from Acme Motor Company
– Q: get $2000 cash back
Q: get $2000 cash back
– R: get a 2% car loan
R: get a 2% car loan
• P
P 
 Q
Q 
 R
R
• Why use XOR here? – example of
Why use XOR here? – example of ambiguity of
ambiguity of
natural languages
natural languages
• When you buy a new car from Acme Motor
When you buy a new car from Acme Motor
car loan.
car loan.

Exercises
Exercises
– P: School is closed
P: School is closed
– Q: 2 feet of snow falls
Q: 2 feet of snow falls
– R: wind chill is below -100
R: wind chill is below -100
• Q
Q 
 R
R 
 P
P
• Precedence among operators:
Precedence among operators:

,
, 
,
, 
,
, 
,
, 

• School is closed if more than 2 feet of snow
School is closed if more than 2 feet of snow

Equivalent Statements
Equivalent Statements
P
P Q
Q 
(P
(P
Q)
Q) (
(
P)
P)
(
(
Q)
Q) 
(P
(P
Q)
Q)
(
(
P)
P)
(
(
Q)
Q)
T
T T
T F
F F
F T
T
T
T F
F T
T T
T T
T
F
F T
T T
T T
T T
T
F
F F
F T
T T
T T
T
The statements
The statements 
(P
(P
Q) and (
Q) and (
P)
P) 
 (
(
Q) are
Q) are logically equivalent
logically equivalent, since they have the
, since they have the
same truth table, or put it in another way,
same truth table, or put it in another way, 
(P
(P
Q)
Q) 
(
(
P)
P) 
 (
(
Q) is always true.
Q) is always true.

Tautologies and Contradictions
Tautologies and Contradictions
A tautology is a statement that is always true.
A tautology is a statement that is always true.
Examples:
Examples:
– R
R
(
(
R)
R)
 
(P
(P
Q)
Q) 
 (
(
P)
P)
(
(
 Q)
Q)
A contradiction is a statement that is always false.
A contradiction is a statement that is always false.
Examples:
Examples:
– R
R
(
(
R)
R)
 
(
(
(P
(P 
 Q)
Q) 
 (
(
P)
P) 
 (
(
Q))
Q))
The negation of any tautology is a contradiction, and
The negation of any tautology is a contradiction, and
the negation of any contradiction is a tautology.
the negation of any contradiction is a tautology.

Equivalence
Equivalence
Definition: two propositional statements
Definition: two propositional statements
S1 and S2 are said to be (logically)
S1 and S2 are said to be (logically)
equivalent, denoted S1
equivalent, denoted S1 
 S2 if
S2 if
– They have the same truth table, or
They have the same truth table, or
– S1
S1 
 S2 is a tautology
S2 is a tautology
Equivalence can be established by
Equivalence can be established by
– Constructing truth tables
Constructing truth tables
– Using equivalence laws (Table 5 in Section 1.2)
Using equivalence laws (Table 5 in Section 1.2)

Equivalence
Equivalence
Equivalence laws
Equivalence laws
– Identity laws,
Identity laws, P
P 
 T
T 
 P,
P,
– Domination laws,
Domination laws, P
P 
 F
F 
 F,
F,
– Idempotent laws,
Idempotent laws, P
P 
 P
P 
 P,
P,
– Double negation law,
Double negation law, 
 (
(
 P
P)
) 
 P
P
– Commutative laws,
Commutative laws, P
P 
 Q
Q 
 Q
Q 
 P,
P,
– Associative laws,
Associative laws, P
P 
 (Q
(Q 
 R)
R)
 (
(P
P 
 Q)
Q) 
 R
R,
,
– Distributive laws,
Distributive laws, P
P 
 (Q
(Q 
 R)
R)
 (
(P
P 
 Q)
Q) 
 (P
(P 
 R)
R),
,
– De Morgan’s laws,
De Morgan’s laws, 
 (P
(P
Q)
Q) 
 (
(
 P)
P) 
 (
(
 Q)
Q)
– Law with implication
Law with implication P
P 
 Q
Q 
 
 P
P 
 Q
Q

Exercises
Exercises
• Show that
Show that P
P 
 Q
Q 
 
 P
P 
 Q
Q: by truth table
: by truth table
• Show that
Show that (P
(P 
 Q)
Q) 
 (P
(P 
 R)
R) 
 P
P 
 (Q
(Q 
 R)
R): by
: by
equivalence laws (q20, p27):
equivalence laws (q20, p27):
– Law with implication on both sides
Law with implication on both sides
– Distribution law on LHS
Distribution law on LHS

Summary, Sections 1.1, 1.2
•Proposition
Proposition
– Statement, Truth value,
Statement, Truth value,
– Proposition, Propositional symbol, Open proposition
Proposition, Propositional symbol, Open proposition
•Operators
Operators
– Define by truth tables
Define by truth tables
– Composite propositions
Composite propositions
– Tautology and contradiction
Tautology and contradiction
•Equivalence of propositional statements
Equivalence of propositional statements
– Definition
Definition
– Proving equivalence (by truth table or equivalence
Proving equivalence (by truth table or equivalence
laws)
laws)

Propositional Functions & Predicates
Propositional Functions & Predicates
Propositional function (open sentence):
Propositional function (open sentence):
statement involving one or more variables,
statement involving one or more variables,
e.g.: x-3 > 5.
e.g.: x-3 > 5.
Let us call this propositional function P(x), where
Let us call this propositional function P(x), where
P is the
P is the predicate
predicate and x is the
and x is the variable
variable.
.
What is the truth value of P(2) ?
What is the truth value of P(2) ? false
false
false
false
true
true
When a variable is given a value, it is said to be
When a variable is given a value, it is said to be
instantiated
instantiated
Truth value depends on value of variable
Truth value depends on value of variable

Propositional Functions
Let us consider the propositional function
Let us consider the propositional function
Q(x, y, z) defined as:
Q(x, y, z) defined as:
x + y = z.
x + y = z.
Here, Q is the
Here, Q is the predicate
predicate and x, y, and z are the
and x, y, and z are the
variables
variables.
.
What is the truth value of Q(2, 3, 5) ?
What is the truth value of Q(2, 3, 5) ? true
true
What is the truth value of Q(9, -9, 0) ?
What is the truth value of Q(9, -9, 0) ?
false
false
true
true
A propositional function (predicate) becomes a
A propositional function (predicate) becomes a
proposition when
proposition when all
all its variables are
its variables are instantiated
instantiated.
.

Other examples of propositional functions
Other examples of propositional functions
Person(x),
Person(x), which is true if x is a person
which is true if x is a person
Person(Socrates) = T
Person(Socrates) = T
CSCourse(x),
CSCourse(x), which is true if x is a
which is true if x is a
computer science course
computer science course
CSCourse(CMSC201) = T
CSCourse(CMSC201) = T
Person(dolly-the-sheep) = F
Person(dolly-the-sheep) = F
CSCourse(MATH155) = F
CSCourse(MATH155) = F
How do we say
How do we say
All humans are mortal
All humans are mortal
One CS course
One CS course

Universal Quantification
Let P(x) be a predicate (propositional function).
Let P(x) be a predicate (propositional function).
Universally quantified sentence
Universally quantified sentence:
:
For all x in the
For all x in the universe of discourse
universe of discourse P(x) is true.
P(x) is true.
Using the universal quantifier
Using the universal quantifier 
:
:

x P(x)
x P(x) “for all x P(x)” or “for every x P(x)”
“for all x P(x)” or “for every x P(x)”
(Note:
(Note: 
x P(x) is either true or false, so it is a
x P(x) is either true or false, so it is a
proposition, not a propositional function.)
proposition, not a propositional function.)

Example: Let the universe of discourse be all people
Example: Let the universe of discourse be all people
S(x): x is a UMBC student.
S(x): x is a UMBC student.
G(x): x is a genius.
What does
What does 
x (S(x)
x (S(x) 
 G(x))
G(x)) mean ?
mean ?
“
“If x is a UMBC student, then x is a genius.” or
If x is a UMBC student, then x is a genius.” or
“
“All UMBC students are geniuses.”
All UMBC students are geniuses.”
If the universe of discourse is all UMBC students,
If the universe of discourse is all UMBC students,
then the same statement can be written as
then the same statement can be written as

x G(x)
x G(x)

Existential Quantification
Existentially quantified sentence
Existentially quantified sentence:
:
There exists an x in the universe of discourse
There exists an x in the universe of discourse
for which P(x) is true.
for which P(x) is true.
Using the existential quantifier
Using the existential quantifier 
:
:

x P(x)
x P(x) “There is an x such that P(x).”
“There is an x such that P(x).”
“
“There is at least one x such that P(x).”
There is at least one x such that P(x).”
(Note:
(Note: 
x P(x) is either true or false, so it is a
x P(x) is either true or false, so it is a
proposition, but no propositional function.)
proposition, but no propositional function.)

Example:
Example:
P(x): x is a UMBC professor.
P(x): x is a UMBC professor.
What does
What does 
x (P(x)
x (P(x) 
 G(x))
G(x)) mean ?
mean ?
“
“There is an x such that x is a UMBC professor
There is an x such that x is a UMBC professor
and x is a genius.”
and x is a genius.”
or
or
“
“At least one UMBC professor is a genius.”
At least one UMBC professor is a genius.”

Quantification
Quantification
Another example:
Another example:
Let the universe of discourse be the real numbers.
Let the universe of discourse be the real numbers.
What does
What does 
x
x
y (x + y = 320)
y (x + y = 320) mean ?
mean ?
“
“For every x there exists a y so that x + y = 320.”
For every x there exists a y so that x + y = 320.”
Is it true?
Is it true?
Is it true for the natural numbers?
Is it true for the natural numbers?
yes
yes
no
no

Disproof by Counterexample
Disproof by Counterexample
A counterexample to
A counterexample to 
x P(x) is an object c so
x P(x) is an object c so
that P(c) is false.
that P(c) is false.
Statements such as
Statements such as 
x (P(x)
x (P(x) 
 Q(x)) can be
Q(x)) can be
disproved by simply providing a counterexample.
disproved by simply providing a counterexample.
Statement: “All birds can fly.”
Statement: “All birds can fly.”
Disproved by counterexample: Penguin.
Disproved by counterexample: Penguin.

Negation
Negation

(
(
x P(x)) is logically equivalent to
x P(x)) is logically equivalent to 
x (
x (
P(x)).
P(x)).

(
(
x P(x)) is logically equivalent to
x P(x)) is logically equivalent to 
x (
x (
P(x)).
P(x)).
See Table 2 in Section 1.3.
See Table 2 in Section 1.3.
This is de Morgan’s law for quantifiers
This is de Morgan’s law for quantifiers

Negation
Negation
Examples
Examples
Not all roses are red
Not all roses are red


x (Rose(x)
x (Rose(x) 
 Red(x)
Red(x))
)

x (Rose(x)
x (Rose(x) 
 
Red(x)
Red(x))
)
Nobody is perfect
Nobody is perfect


x (Person(x)
x (Person(x) 
 Perfect(x)
Perfect(x))
)

x (Person(x)
x (Person(x) 
 
Perfect(x)
Perfect(x))
)

Nested Quantifier
Nested Quantifier
A predicate can have more than one variables.
A predicate can have more than one variables.
– S(x, y, z): z is the sum of x and y
S(x, y, z): z is the sum of x and y
– F(x, y): x and y are friends
F(x, y): x and y are friends
We can quantify individual variables in different
We can quantify individual variables in different
ways
ways
 
x, y, z (S(x, y, z)
x, y, z (S(x, y, z) 
 (x <= z
(x <= z 
 y <= z))
y <= z))
 
x
x 
y
y 
z (F(x, y)
z (F(x, y) 
 F(x, z)
F(x, z) 
 (y != z)
(y != z) 
 
F(y, z)
F(y, z)

Nested Quantifier
Nested Quantifier
Exercise: translate the following English
Exercise: translate the following English
sentence into logical expression
sentence into logical expression
“
“There is a rational number in between every
There is a rational number in between every
pair of distinct rational numbers”
pair of distinct rational numbers”
Use predicate
Use predicate Q(x)
Q(x), which is true when x
, which is true when x
is a rational number
is a rational number

x,y (
x,y (Q
Q(x)
(x) 
 Q
Q (y)
(y) 
 (x < y)
(x < y) 


u (Q(u)
u (Q(u) 
 (x < u)
(x < u) 
 (u < y)))
(u < y)))

• Propositional functions (predicates)
Propositional functions (predicates)
• Universal and existential quantifiers, and
Universal and existential quantifiers, and
the duality of the two
the duality of the two
• When predicates become propositions
When predicates become propositions
– All of its variables are instantiated
All of its variables are instantiated
– All of its variables are quantified
All of its variables are quantified
• Nested quantifiers
Nested quantifiers
– Quantifiers with negation
Quantifiers with negation
• Logical expressions formed by predicates,
Logical expressions formed by predicates,
operators, and quantifiers
operators, and quantifiers

Let’s proceed to…
Let’s proceed to…
Mathematical
Mathematical
Reasoning
Reasoning

Mathematical Reasoning
Mathematical Reasoning
We need
We need mathematical reasoning
mathematical reasoning to
to
• determine whether a mathematical argument is
determine whether a mathematical argument is
correct or incorrect and
correct or incorrect and
• construct mathematical arguments.
construct mathematical arguments.
Mathematical reasoning is not only important for
Mathematical reasoning is not only important for
conducting
conducting proofs
proofs and
and program verification
program verification, but
, but
also for
also for artificial intelligence
artificial intelligence systems (drawing
systems (drawing
logical inferences from knowledge and facts).
logical inferences from knowledge and facts).
We focus on
We focus on deductive
deductive proofs
proofs

Terminology
Terminology
An
An axiom
axiom is a basic assumption about mathematical
is a basic assumption about mathematical
structure that needs no proof.
structure that needs no proof.
- Things known to be true (facts or proven theorems)
Things known to be true (facts or proven theorems)
- Things believed to be true but cannot be proved
Things believed to be true but cannot be proved
We can use a
We can use a proof
proof to demonstrate that a
to demonstrate that a
particular statement is true. A proof consists of a
particular statement is true. A proof consists of a
sequence of statements that form an argument.
sequence of statements that form an argument.
The steps that connect the statements in such a
The steps that connect the statements in such a
sequence are the
sequence are the rules of inference
rules of inference.
.
Cases of incorrect reasoning are called
Cases of incorrect reasoning are called fallacies
fallacies.
.

Terminology
Terminology
A
A theorem
theorem is a statement that can be shown to be
is a statement that can be shown to be
true.
true.
A
A lemma
lemma is a simple theorem used as an
is a simple theorem used as an
intermediate result in the proof of another
intermediate result in the proof of another
theorem.
theorem.
A
A corollary
corollary is a proposition that follows directly
is a proposition that follows directly
from a theorem that has been proved.
from a theorem that has been proved.
A
A conjecture
conjecture is a statement whose truth value is
is a statement whose truth value is
unknown. Once it is proven, it becomes a theorem.
unknown. Once it is proven, it becomes a theorem.

Proofs
Proofs
A
A theorem
theorem often has two parts
often has two parts
- Conditions (premises, hypotheses)
Conditions (premises, hypotheses)
- conclusion
conclusion
A
A correct (deductive) proof
correct (deductive) proof is to establish that
is to establish that
- If the conditions are true then the conclusion is true
If the conditions are true then the conclusion is true
- I.e., Conditions
I.e., Conditions 
 conclusion is a
conclusion is a tautology
tautology
Often there are missing pieces between conditions
Often there are missing pieces between conditions
and conclusion. Fill it by an
and conclusion. Fill it by an argument
argument
- Using conditions and axioms
Using conditions and axioms
- Statements in the argument connected by proper
Statements in the argument connected by proper
rules of inference
rules of inference

Rules of Inference
Rules of Inference
Rules of inference
Rules of inference provide the justification of
provide the justification of
the steps used in a proof.
the steps used in a proof.
One important rule is called
One important rule is called modus ponens
modus ponens or the
or the
law of detachment
law of detachment. It is based on the tautology
. It is based on the tautology
(p
(p 
 (p
(p 
 q))
q)) 
 q. We write it in the following way:
q. We write it in the following way:
p
p
p
p 
 q
q
____
____

 q
q
The two
The two hypotheses
hypotheses p and p
p and p 
 q are
q are
written in a column, and the
written in a column, and the conclusion
conclusion
below a bar, where
below a bar, where 
 means “therefore”.
means “therefore”.

Rules of Inference
Rules of Inference
The general form of a rule of inference is:
The general form of a rule of inference is:
p
p1
1
p
p2
2
.
.
.
.
.
.
p
pn
n
____
____

 q
q
The rule states that if p
The rule states that if p1
1 and
and p
p2
2 and
and …
…
and
and p
pn
n are all true, then q is true as well.
are all true, then q is true as well.
Each rule is an established tautology of
Each rule is an established tautology of
p
p1
1 
 p
p2
2 
 …
… 
 p
pn
n 
 q
q
These rules of inference can be used in
These rules of inference can be used in
any mathematical argument and do not
any mathematical argument and do not
require any proof.
require any proof.

Rules of Inference
Rules of Inference
p
p
_____
_____

 p
p
q
q
Addition
Addition
p
p
q
q
_____
_____

 p
p
Simplification
Simplification
p
p
q
q
_____
_____

 p
p
q
q
Conjunction
Conjunction

q
q
p
p 
 q
q
_____
_____

 
 p
p
Modus
Modus
tollens
tollens
p
p 
 q
q
q
q 
 r
r
_____
_____

 p
p
 r
r
Hypothetical
Hypothetical
syllogism
syllogism
(
(chaining
chaining)
)
p
p
q
q

p
p
_____
_____

 q
q
Disjunctive
Disjunctive
syllogism
syllogism
(
(resolution
resolution)
)

Arguments
Arguments
Just like a rule of inference, an
Just like a rule of inference, an argument
argument consists
consists
of one or more hypotheses (or premises) and a
of one or more hypotheses (or premises) and a
conclusion.
conclusion.
We say that an argument is
We say that an argument is valid
valid, if whenever all
, if whenever all
its hypotheses are true, its conclusion is also true.
its hypotheses are true, its conclusion is also true.
However, if any hypothesis is false, even a valid
However, if any hypothesis is false, even a valid
argument can lead to an incorrect conclusion.
argument can lead to an incorrect conclusion.
Proof: show that
Proof: show that hypotheses
hypotheses 
 conclusion
conclusion is true
is true
using rules of inference
using rules of inference

Arguments
Arguments
Example:
Example:
“
“If 101 is divisible by 3, then 101
If 101 is divisible by 3, then 1012
2
is divisible by 9.
is divisible by 9.
101 is divisible by 3. Consequently, 101
101 is divisible by 3. Consequently, 1012
2
is divisible
is divisible
by 9.”
by 9.”
Although the argument is
Although the argument is valid
valid, its conclusion is
, its conclusion is
incorrect
incorrect, because one of the hypotheses is false
, because one of the hypotheses is false
(“101 is divisible by 3.”).
(“101 is divisible by 3.”).
If in the above argument we replace 101 with 102,
If in the above argument we replace 101 with 102,
we could correctly conclude that 102
we could correctly conclude that 1022
2
is divisible
is divisible
by 9.
by 9.

Arguments
Arguments
Which rule of inference was used in the last
Which rule of inference was used in the last
argument?
argument?
p: “101 is divisible by 3.”
p: “101 is divisible by 3.”
q: “101
q: “1012
2
is divisible by 9.”
is divisible by 9.”
p
p
p
p 
 q
q
_____
_____

 q
q
Modus
Modus
ponens
ponens
Unfortunately, one of the hypotheses (p) is false.
Unfortunately, one of the hypotheses (p) is false.
Therefore, the conclusion q is incorrect.
Therefore, the conclusion q is incorrect.

Arguments
Arguments
Another example:
Another example:
“
“If it rains today, then we will not have a barbeque
If it rains today, then we will not have a barbeque
today. If we do not have a barbeque today, then
today. If we do not have a barbeque today, then
we will have a barbeque tomorrow.
we will have a barbeque tomorrow.
Therefore, if it rains today, then we will have a
Therefore, if it rains today, then we will have a
barbeque tomorrow.”
barbeque tomorrow.”
This is a
This is a valid
valid argument: If its hypotheses are
argument: If its hypotheses are
true, then its conclusion is also true.
true, then its conclusion is also true.

Arguments
Arguments
Let us formalize the previous argument:
Let us formalize the previous argument:
p: “It is raining today.”
p: “It is raining today.”
q: “We will not have a barbecue today.”
q: “We will not have a barbecue today.”
r: “We will have a barbecue tomorrow.”
r: “We will have a barbecue tomorrow.”
So the argument is of the following form:
So the argument is of the following form:
p
p 
 q
q
q
q 
 r
r
______
______

 P
P 
 r
r
Hypothetical
Hypothetical
syllogism
syllogism

Arguments
Arguments
Another example:
Another example:
Gary is either intelligent or a good actor.
Gary is either intelligent or a good actor.
If Gary is intelligent, then he can count
If Gary is intelligent, then he can count
from 1 to 10.
from 1 to 10.
Gary can only count from 1 to 3.
Therefore, Gary is a good actor.
Therefore, Gary is a good actor.
i: “Gary is intelligent.”
a: “Gary is a good actor.”
c: “Gary can count from 1 to 10.”

Arguments
Arguments
Step 1:
Step 1: 
 c
c Hypothesis
Hypothesis
Step 2:
Step 2: i
i 
 c
c Hypothesis
Hypothesis
Step 3:
Step 3: 
 i
i Modus tollens Steps 1 & 2
Modus tollens Steps 1 & 2
Step 4:
Step 4: a
a 
 i
i Hypothesis
Hypothesis
Step 5:
Step 5: a
a Disjunctive Syllogism
Disjunctive Syllogism
Steps 3 & 4
Steps 3 & 4
Conclusion:
Conclusion: a
a (“Gary is a good actor.”)
(“Gary is a good actor.”)

Arguments
Arguments
Yet another example:
Yet another example:
If you listen to me, you will pass CS 320.
If you listen to me, you will pass CS 320.
You passed CS 320.
You passed CS 320.
Therefore, you have listened to me.
Therefore, you have listened to me.
Is this argument valid?
Is this argument valid?
No
No, it assumes ((p
, it assumes ((p 
 q)
q)
 q)
q) 
 p.
p.
This statement is not a tautology. It is
This statement is not a tautology. It is false
false if p is
if p is
false and q is true.
false and q is true.

Rules of Inference for Quantified Statements

x P(x)
x P(x)
__________
__________

 P(c) if c
P(c) if c
U
U
Universal
Universal
instantiation
instantiation
P(c) for an arbitrary c
P(c) for an arbitrary c
U
U
___________________
___________________

 
x P(x)
x P(x)
Universal
Universal
generalization
generalization

x P(x)
x P(x)
______________________
______________________

 P(c) for some element c
P(c) for some element c
U
U
Existential
Existential
instantiation
instantiation
P(c) for some element c
P(c) for some element c
U
U
____________________
____________________

 
x P(x)
x P(x)
Existential
Existential
generalization
generalization

Example:
Example:
Every UMB student is a genius.
Every UMB student is a genius.
George is a UMB student.
George is a UMB student.
Therefore, George is a genius.
Therefore, George is a genius.
U(x): “x is a UMB student.”
U(x): “x is a UMB student.”
G(x): “x is a genius.”
G(x): “x is a genius.”

The following steps are used in the argument:
The following steps are used in the argument:
Step 1:
Step 1: 
x (U(x)
x (U(x) 
 G(x))
G(x)) Hypothesis
Hypothesis
Step 2:
Step 2: U(George)
U(George) 
 G(George)
G(George) Univ. instantiation
Univ. instantiation
using Step 1
using Step 1

x P(x)
x P(x)
__________
__________

 P(c) if c
P(c) if c
U
U
Universal
Universal
instantiation
instantiation
Step 3:
Step 3: U(George)
U(George) Hypothesis
Hypothesis
Step 4:
Step 4: G(George)
G(George) Modus ponens
Modus ponens
using Steps 2 & 3
using Steps 2 & 3

Proving Theorems
Proving Theorems
Direct proof:
Direct proof:
An implication p
An implication p 
 q can be proved by showing that
q can be proved by showing that
if p is true, then q is also true.
if p is true, then q is also true.
Example:
Example: Give a direct proof of the theorem
Give a direct proof of the theorem
“If n is odd, then n
“If n is odd, then n2
2
is odd.”
is odd.”
Idea:
Idea: Assume that the hypothesis of this
Assume that the hypothesis of this
implication is true (n is odd). Then use rules of
implication is true (n is odd). Then use rules of
inference and known theorems of math to show
inference and known theorems of math to show
that q must also be true (n
that q must also be true (n2
2
is odd).
is odd).

Proving Theorems
Proving Theorems
n is odd.
n is odd.
Then n = 2k + 1, where k is an integer.
Then n = 2k + 1, where k is an integer.
Consequently, n
Consequently, n2
2
= (2k + 1)
= (2k + 1)2
2
.
.
= 4k
= 4k2
2
+ 4k + 1
+ 4k + 1
= 2(2k
= 2(2k2
2
+ 2k) + 1
+ 2k) + 1
Since n
Since n2
2
can be written in this form, it is odd.
can be written in this form, it is odd.

Proving Theorems
Proving Theorems
Indirect proof:
Indirect proof:
An implication p
An implication p 
 q is equivalent to its
q is equivalent to its contra-
contra-
positive
positive 
q
q 
 
p. Therefore, we can prove p
p. Therefore, we can prove p 
 q
q
by showing that whenever q is false, then p is also
by showing that whenever q is false, then p is also
false.
false.
Example:
Example: Give an indirect proof of the theorem
Give an indirect proof of the theorem
“If 3n + 2 is odd, then n is odd.”
“If 3n + 2 is odd, then n is odd.”
Idea:
Idea: Assume that the conclusion of this
Assume that the conclusion of this
implication is false (n is even). Then use rules of
implication is false (n is even). Then use rules of
inference and known theorems to show that p
inference and known theorems to show that p
must also be false (3n + 2 is even).
must also be false (3n + 2 is even).

Proving Theorems
Proving Theorems
n is even.
n is even.
Then n = 2k, where k is an integer.
It follows that 3n + 2 = 3(2k) + 2
= 6k + 2
= 6k + 2
= 2(3k + 1)
= 2(3k + 1)
Therefore, 3n + 2 is even.
We have shown that the contrapositive of the
We have shown that the contrapositive of the
implication is true, so the implication itself is also
implication is true, so the implication itself is also
true
true (If 3n + 2 is odd, then n is odd).
(If 3n + 2 is odd, then n is odd).

Proving Theorems
Proving Theorems
Indirect Proof is a special case of
Indirect Proof is a special case of proof by
proof by
contradiction
contradiction
Suppose n is even (
Suppose n is even (negation of the conclusion
negation of the conclusion).
).
= 6k + 2
= 6k + 2
= 2(3k + 1)
= 2(3k + 1)
However, this is a
However, this is a contradiction
contradiction since 3n + 2 is given
since 3n + 2 is given
to be odd, so the conclusion (n is odd) holds.
to be odd, so the conclusion (n is odd) holds.

Another Example on Proof
Anyone performs well is either intelligent or a
Anyone performs well is either intelligent or a
good actor.
good actor.
If someone is intelligent, then he/she can count
from 1 to 10.
from 1 to 10.
Gary performs well.
Gary performs well.
Therefore, not everyone is both intelligent and a
Therefore, not everyone is both intelligent and a
good actor
good actor
P(x): x performs well
P(x): x performs well
I(x): x is intelligent
I(x): x is intelligent
A(x): x is a good actor
A(x): x is a good actor
C(x): x can count from 1 to 10
C(x): x can count from 1 to 10

Hypotheses:
Hypotheses:
1.
1. Anyone performs well is either intelligent or a good
Anyone performs well is either intelligent or a good
actor.
actor.

x (P(x)
x (P(x) 
 I(x)
I(x) 
 A(x))
A(x))
2.
2. If someone is intelligent, then he/she can count
from 1 to 10.
from 1 to 10.

x (I(x)
x (I(x) 
 C
C(x) )
(x) )
3.
3. Gary performs well.
Gary performs well.
P(G)
P(G)
4.
4. Gary can only count from 1 to 3.

C(G)
C(G)
Conclusion: not everyone is both intelligent and a good
Conclusion: not everyone is both intelligent and a good
actor
actor

x(I(x)
x(I(x) 
 A(x))
A(x))

Direct proof:
Direct proof:
Step 1:
Step 1: 
x (P(x)
x (P(x) 
 I(x)
I(x) 
 A(x))
A(x)) Hypothesis
Hypothesis
Step 2:
Step 2: P(G)
P(G) 
 I(G)
I(G) 
 A(G)
A(G) Univ. Inst. Step 1
Univ. Inst. Step 1
Step 3:
Step 3: P(G)
P(G) Hypothesis
Hypothesis
Step 4:
Step 4: I(G)
I(G) 
 A(G)
A(G) Modus ponens Steps 2 & 3
Modus ponens Steps 2 & 3
Step 5:
Step 5: 
x (I(x)
x (I(x) 
 C(x))
C(x)) Hypothesis
Hypothesis
Step 6:
Step 6: I(G)
I(G) 
 C(G)
C(G) Univ. inst. Step5
Univ. inst. Step5
Step 7:
Step 7: 
C(G)
C(G) Hypothesis
Hypothesis
Step 8:
Step 8: 
I(G)
I(G) Modus tollens Steps 6 & 7
Modus tollens Steps 6 & 7
Step 9:
Step 9: 
I(G)
I(G) 
 
A(G)
A(G) Addition Step 8
Addition Step 8
Step 10:
Step 10: 
(I(G)
(I(G) 
 A(G))
A(G)) Equivalence Step 9
Equivalence Step 9
Step 11:
Step 11: 
x
x
(I(x)
(I(x) 
 A(x))
A(x)) Exist. general. Step 10
Exist. general. Step 10
Step 12:
Step 12: 
x (I(x)
x (I(x) 
 A(x))
A(x)) Equivalence Step 11
Equivalence Step 11
Conclusion:
Conclusion: 
x (I(x)
x (I(x) 
 A(x))
A(x)), not everyone is both intelligent and
, not everyone is both intelligent and
a good actor.
a good actor.

Summary, Section 1.5
Summary, Section 1.5
• Terminology (axiom, theorem, conjecture,
Terminology (axiom, theorem, conjecture,
argument, etc.)
argument, etc.)
• Rules of inference (Tables 1 and 2)
Rules of inference (Tables 1 and 2)
• Valid argument (hypotheses and conclusion)
Valid argument (hypotheses and conclusion)
• Construction of valid argument using rules of
Construction of valid argument using rules of
inference
inference
– For each rule used, write down and the statements
For each rule used, write down and the statements
involved in the proof
involved in the proof
• Direct and indirect proofs
Direct and indirect proofs
– Other proof methods (e.g., induction, pigeon hole)
Other proof methods (e.g., induction, pigeon hole)
will be introduced in later chapters
will be introduced in later chapters

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