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Discrete Mathematics
Lecture 2.
Dr.Bassant Mohamed El-Bagoury
[email protected]
Module
Logic (part 2 --- proof methods)
1
Outline
1. Mathematical Reasoning
2. Arguments Examples – Predicate Logic
3. Rules of Inference – Knowledge
Engineering
4. Rules of Inference for Quantifiers
4. Methods for Theorem Proving
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Mathematical Reasoning
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Mathematical Reasoning
We need mathematical reasoning to
• determine whether a mathematical argument is
correct or incorrect and
• construct mathematical arguments.
Mathematical reasoning is not only important for
conducting proofs and program verification, but
also for artificial intelligence systems (drawing
inferences).
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Arguments
Example:
Gary is either intelligent or a good actor.
If Gary is intelligent, then he can count
from 1 to 10.
Gary can only count from 1 to 2.
Therefore, Gary is a good actor.
i: “Gary is intelligent.”
a: “Gary is a good actor.”
c: “Gary can count from 1 to 10.”
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Arguments
i: “Gary is intelligent.”
a: “Gary is a good actor.”
c: “Gary can count from 1 to 10.”
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
c
ic
i
ai
a
Hypothesis
Hypothesis
Modus Tollens Steps 1 & 2
Hypothesis
Disjunctive Syllogism
Steps 3 & 4
Conclusion: a (“Gary is a good actor.”)
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Arguments
Another example:
“If it rains today, then we will not have a barbeque
today. If we do not have a barbeque today, then we
will have a barbeque tomorrow.
Therefore, if it rains today, then we will have a
barbeque tomorrow.”
This is a valid argument: If its hypotheses are true,
then its conclusion is also true.
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AI research × Software
Engineering
Requirements
Specification
Intelligent
Behaviour
Knowledge Level
System Level
Symbol Level
Functional
Specification
System
Implementation
A small Expert System ( Inference System ) of rules for analysis of
automotive problems.
Rule 1:
if
the engine is getting gas, and
the engine will turn over,
then
the problem is spark plugs.
Rule 2:
if
the engine does not turn over, and
the lights do not come on
then
the problem is battery or cables.
Rule 3:
if
the engine does not turn over, and
the lights do come on
then
the problem is the starter motor.
Rule 4:
if
there is gas in the fuel tank, and
there is gas in the carburetor
then
the engine is getting gas.
Fig
The Inference System at the start of a consultation in the car
diagnostic example.
Fig
The Inference System after Rule 1 has fired.
The and/or graph searched in the car diagnosis example, with the
conclusion of Inference Rule 4 matching the first premise of Inference
Rule 1.
The following dialogue begins with the computer asking the user about
the goals present in working memory.
Gas in fuel tank?
Yes
Gas in carburetor?
Yes
Engine will turn over?
Why
It has been established that:
1. The engine is getting gas,
2. The engine will turn over,
Then the problem is the spark plugs.
How the engine is getting gas
This follows from rule 4:
if
gas in fuel tank, and
gas in carburetor
then
engine is getting gas.
gas in fuel tank was given by the user
gas in carburetor was given by the user
Terminology
An axiom is a basic assumption about mathematical
structures that needs no proof.
We can use a proof to demonstrate that a particular
statement is true. A proof consists of a sequence of
statements that form an argument.
The steps that connect the statements in such a
sequence are the rules of inference.
Cases of incorrect reasoning are called fallacies.
A theorem is a statement that can be shown to be
true.
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Terminology
A lemma is a simple theorem used as an
intermediate result in the proof of another theorem.
A corollary is a proposition that follows directly from
a theorem that has been proved.
A conjecture is a statement whose truth value is
unknown. Once it is proven, it becomes a theorem.
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Arguments
Just like a rule of inference, an argument consists
of one or more hypotheses and a conclusion.
We say that an argument is valid, if whenever all its
hypotheses are true, its conclusion is also true.
However, if any hypothesis is false, even a valid
argument can lead to an incorrect conclusion.
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Arguments
Example:
“If 101 is divisible by 3, then 1012 is divisible by 9.
101 is divisible by 3. Consequently, 1012 is divisible
by 9.”
Although the argument is valid, its conclusion is
incorrect, because one of the hypotheses is false
(“101 is divisible by 3.”).
If in the above argument we replace 101 with 102,
we could correctly conclude that 1022 is divisible by
9.
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Theorems, proofs, and rules of inference
When is a mathematical argument (or “proof”) correct?
What techniques can we use to construct a mathematical
argument?
Theorem – statement that can be shown to be true.
Axioms or postulates or premises – statements which are given
and assumed to be true.
Proof – sequence of statements, a valid Argument, to show
that a theorem is true.
Rules of Inference – rules used in a proof to draw
conclusions from assertions known to be true.
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Valid Arguments
(reminder)
Recall:
An argument is a sequence of propositions. The final proposition is called the
conclusion of the argument while the other propositions are called the
premises or hypotheses of the argument.
An Argument is valid whenever the truth of all its premises implies the truth
of its conclusion.
How to show that q logically follows from the hypotheses (p1  p2  …pn)?
Show that
(p1  p2  …pn)  q is a tautology
One can use the rules of inference to show the validity of an argument.
Vacuous proof - if one of the premises is false then (p1  p2  …pn)  q
is vacuously True, since False implies anything.
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Methods for Proving Theorems
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Methods of Proof
1) Direct Proof
2) Proof by Contraposition
3) Proof by Contradiction
4) Proof of Equivalences
5) Proof by Cases
6) Existence Proofs
7) Counterexamples
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1) Direct Proof
Proof statement :
pq
by:
Assume p
From p derive q.
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Direct proof --- Example 1
Here’s what you know:
Mary is a Math major or a CS major.
If Mary does not like discrete math, she is not a CS major.
If Mary likes discrete math, she is smart.
Let
Mary is not a math major.
M - Mary is a Math major
Can you conclude Mary is smart?
C – Mary is a CS major
Informally, what’s the inference chain of reasoning? D – Mary likes discrete math
S – Mary is smart
MC
D  C
D 
S C)  (D  S)  (M))  S
((M  C)  (D
M
?
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In general, to prove p  q, assume p and show that q follows.
((M  C)  (D  C)  (D  S)  (M))  S
?
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See Table 1, p. 66, Rosen.
Reminder: Propositional logic
Rules of Inference or Method of Proof
Rule of Inference
Tautology (Deduction Theorem)
Name
P
PQ
P  (P  Q)
Addition
PQ
P
(P  Q)  P
Simplification
P
Q
PQ
[(P)  (Q)]  (P  Q)
Conjunction
P
PQ
Q
[(P)  (P Q)]  P
Modus Ponens
Q
PQ
 P
[(Q)  (P Q)]  P
Modus Tollens
PQ
QR
 P R
[(PQ)  (Q  R)]  (PR)
Hypothetical Syllogism
(“chaining”)
PQ
P
Q
[(P  Q)  (P)]  Q
Disjunctive syllogism
PQ
P  R
QR
[(P  Q)  (P  R)]  (Q  R)
Resolution
Subsumes MP
Example 1 - direct proof
1.
2.
3.
4.
5.
6.
7.
MC
D  C
DS
M
C
D
S
Given (premise)
Given
Given
Given
DS (disjunctive syllogism; 1,4)
MT (modus tollens; 2,5)
MP (modus ponens; 3,6)
QED
Mary is smart!
QED or Q.E.D. --- quod erat demonstrandum
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Direct Proof --- Example 2
Theorem:
If n is odd integer, then n2 is odd.
Looks plausible, but…
How do we proceed? How do we prove this?
Start with
Definition: An integer is even if there exists an integer k such that n = 2k,
and n is odd if there exists an integer k such that n = 2k+1.
Properties: An integer is even or odd; and no integer is
both even and odd. (aside: would require proof.)
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Example 2: Direct Proof
Theorem:
(n) P(n)  Q(n),
where P(n) is “n is an odd integer” and Q(n) is “n2 is odd.”
We will show P(n)  Q(n)
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Theorem:
If n is odd integer, then n2 is odd.
Proof:
Let P --- “n is odd integer”
Q --- “n2 is odd”
we want to show that P  Q
• Assume P, i.e., n is odd.
• By definition n = 2k + 1, where k is some integer.
• Therefore n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2 (2k2 + 2k ) + 1,
which is by definition is an odd number (use k’ = (2k2 + 2k ) ).
QED
Proof strategy hint: Go back to definitions of concepts
and start by trying direct proof.
MORE EXPLAINATION
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The Foundations: Logic and Proofs
Chapter 1
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Propositional Logic
Proposition is a declarative statement that
is either true of false
•Baton Rouge is the capital of Louisiana
•Toronto is the capital of Canada
•1+1=2
•2+2=3
True
False
True
False
Statements which are not propositions:
•What time is it?
•x+1 = 2
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p  today is Thursday
Negation: p  today is not Thursday
truth table
p
p
T
F
F
T
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p  today is Thursday
q  it is raining today
Conjunction:
p  q  today is Thursday and it is raining today
truth table
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F
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p  today is Thursday
q  today is Friday
Disjunction:
p  q  today is Thursday or today is Friday
truth table
p
q
pq
T
T
T
T
F
T
F
T
T
F
F
F
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p  today is Thursday
q  today is Friday
Exclusive-or: one or the other but not both
p  q  today is Thursday or today is Friday (but not both)
truth table
p
q
pq
T
T
F
T
F
T
F
T
T
F
F
F
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(hypothesis)
p  Maria learns discrete math
(conclusion)
q  Maria will find a good job
Conditional statement:
p  q  if Maria learns discrete math then she will find a good job
if p then q
p implies q
q follows from p
p only if q
p is sufficient for q
truth table
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
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Conditional statement:
pq
equivalent
Contrapositive: q
Converse:
Inverse:
 p
(same
truth table)
q p
p  q
equivalent
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p  you can take the flight
q  you buy a ticket
Biconditional statement:
p  q  you can take the flight if and only if you buy a ticket
p if and only if q
p iff q
If p then q and conversely
p is necessary and sufficient for q
truth table
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
T
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Compound propositions
p
q
q
p  q
p  q p  q  p  q
T
T
F
T
T
T
T
F
T
T
F
F
F
T
F
F
F
T
F
F
T
T
F
F
Precedence of operators
    
higher
lower
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Translating English into propositions
p " you cannot ride the roller coaster if you are under 4 feet tall
unless you are older than 16 years old"
q  you can ride the roller coaster
r  you are under 4 feet tall
s  you are older than 16 years old
p  r  s  q
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Propositional Equivalences
Compound proposition
Tautology: always true
Contradiction: always false
tautology
p  p
contradiction
p
p
p  p
T
F
T
F
F
T
T
F
Contingency: not a tautology and
not a contradiction
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Rules of Inference
If you have a current password,
then you can log onto the network
You have a current password
p
Therefore,
you can log onto the network
q
Valid argument:
if premises are true
then conclusion is true
pq
Modus Ponens
pq
p
q
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Modus Ponens
pq
p
q
(( p  q)  p)  q
If p  q and p then q
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Rules of Inference
Modus Ponens
Modus Tollens
pq
pq
p
q
q
 p
Hypothetical
Syllogism
pq
Disjunctive
Syllogism
pq
qr
p
pr
q
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Rules of Inference
Addition
Simplification
p
pq
Conjunction
pq
p
Resolution
q
pq
p  r
pq
q  r
p
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It is below freezing now
Therefore,
p it is either below freezing
q or raining now
p
pq
Addition
p
pq
47
p It is below freezing
q and raining now
Therefore,
it is below freezing now
pq
p
Simplification
pq
p
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p
q
If it rains today
then we will not have a barbecue today
q
r
If we do not have a barbecue today
then we will have a barbecue tomorrow
p
r
Therefore,
if it rains today
then we will have a barbecue tomorrow
Hypothetical
Syllogism
pq
qr
pr
pq
qr
pr
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p
q
p
r
it is not snowing
or Jasmine is skiing
p  q
It is snowing
or Bart is playing hockey
pr
Therefore,
Jasmine is skiing
or Bart is playing hockey
qr
Resolution
pq
p  r
q  r
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Hypothesis:
p
q
It is not sunny this afternoon
and it is colder than yesterday
r We will go swimming
p only if it is sunny
r
s
s
t
If we do not go swimming,
then we will take a canoe trip
If we take a canoe trip,
then we will be home by sunset
p  q
rp
r  s
s t
Conclusion:
t
We will be home by sunset
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1.  p  q
Hypothesis
2.  p
Simplification from 1
3. r  p
Hypothesis
4.  r
Modus tollens from 2,3
5.  r  s
Hypothesis
6. s
7. s  t
Modus ponens from 4,5
Hypothesis
8. t
Modus ponens from 6,7
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Chapter 1:
Foundations: Logic and Proofs
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Foundations of Logic
(§1.1-1.3)
Mathematical Logic is a tool for working with
complicated compound statements. It includes:
• A language for expressing them.
• A concise notation for writing them.
• A methodology for objectively reasoning about
their truth or falsity.
• It is the foundation for expressing formal proofs in
all branches of mathematics.
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Universes of Discourse (U.D.s)
• The power of distinguishing objects from
predicates is that it lets you state things
about many objects at once.
• E.g., let P(x)=“x+1>x”. We can then say,
“For any number x, P(x) is true” instead of
(0+1>0)  (1+1>1)  (2+1>2)  ...
• The collection of values that a variable x
can take is called x’s universe of discourse.
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Quantifier Expressions
• Quantifiers provide a notation that allows
us to quantify (count) how many objects in
the univ. of disc. satisfy a given predicate.
• “” is the FORLL or universal quantifier.
x P(x) means for all x in the u.d., P holds.
• “” is the XISTS or existential quantifier.
x P(x) means there exists an x in the u.d.
(that is, 1 or more) such that P(x) is true.
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The Universal Quantifier 
• Example:
Let the u.d. of x be parking spaces at UF.
Let P(x) be the predicate “x is full.”
Then the universal quantification of P(x),
x P(x), is the proposition:
– “All parking spaces at UF are full.”
– i.e., “Every parking space at UF is full.”
– i.e., “For each parking space at UF, that space is full.”
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The Existential Quantifier 
• Example:
Let the u.d. of x be parking spaces at UF.
Let P(x) be the predicate “x is full.”
Then the existential quantification of P(x),
x P(x), is the proposition:
– “Some parking space at UF is full.”
– “There is a parking space at UF that is full.”
– “At least one parking space at UF is full.”
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Review: Predicate Logic (§1.3)
• Objects x, y, z, …
• Predicates P, Q, R, … are functions
mapping objects x to propositions P(x).
• Multi-argument predicates P(x, y).
• Quantifiers: [x P(x)] :≡ “For all x’s, P(x).”
[x P(x)] :≡ “There is an x such that P(x).”
• Universes of discourse, bound & free vars.
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Foundations of Logic: Overview
• Propositional logic (§1.1-1.2):
– Basic definitions. (§1.1)
– Equivalence rules & derivations. (§1.2)
• Predicate logic (§1.3-1.4)
– Predicates.
– Quantified predicate expressions.
– Equivalences & derivations.
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Propositional Logic (§1.1)
Propositional Logic is the logic of compound
statements built from simpler statements
using so-called Boolean connectives.
Some applications in computer science:
• Design of digital electronic circuits.
• Expressing conditions in programs.
• Queries to databases & search engines.
George Boole
(1815-1864)
Chrysippus of Soli
(ca. 281 B.C. – 205 B.C.)
61
Definition of a Proposition
A proposition (p, q, r, …) is simply a statement (i.e.,
a declarative sentence) with a definite meaning,
having a truth value that’s either true (T) or false
(F) (never both, neither, or somewhere in
between).
(However, you might not know the actual truth
value, and it might be situation-dependent.)
[Later we will study probability theory, in which we assign
degrees of certainty to propositions. But for now: think
True/False only!]
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Examples of Propositions
• “It is raining.” (In a given situation.)
• “Beijing is the capital of China.” • “1 + 2 = 3”
But, the following are NOT propositions:
• “Who’s there?” (interrogative, question)
• “La la la la la.” (meaningless interjection)
• “Just do it!” (imperative, command)
• “Yeah, I sorta dunno, whatever...” (vague)
• “1 + 2” (expression with a non-true/false value)
63
Operators / Connectives
An operator or connective combines one or
more operand expressions into a larger
expression. (E.g., “+” in numeric exprs.)
Unary operators take 1 operand (e.g., −3);
Binary operators take 2 operands (eg 3  4).
Propositional or Boolean operators operate on
propositions or truth values instead of on
numbers.
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Some Popular Boolean Operators
Formal Name
Nickname Arity
Symbol
Negation operator
NOT
Unary
¬
Conjunction operator
AND
Binary

Disjunction operator
OR
Binary

Exclusive-OR operator XOR
Binary

Implication operator
IMPLIES
Binary
Biconditional operator
IFF
Binary

↔
65
The Negation Operator
The unary negation operator “¬” (NOT)
transforms a prop. into its logical negation.
E.g. If p = “I have brown hair.”
then ¬p = “I do not have brown hair.”
Truth table for NOT:
p p
T
T :≡ True; F :≡ False
“:≡” means “is defined as”
F
Operand
column
Result
column
66
The Conjunction Operator
The binary conjunction operator “” (AND)
combines two propositions to form their
ND
logical conjunction.
E.g. If p=“I will have salad for lunch.” and
q=“I will have steak for dinner.”, then
pq=“I will have salad for lunch and
I will have steak for dinner.”
Remember: “” points up like an “A”, and it means “ND”
67
Conjunction Truth Table
Operand columns
• Note that a
p q
pq
conjunction
F F
p1  p2  …  pn
F T
of n propositions
T F
will have 2n rows
in its truth table.
T T
• Also: ¬ and  operations together are sufficient to express any Boolean truth table!
68
The Disjunction Operator
• The binary disjunction operator “” (OR)
combines two propositions to form their
logical disjunction.
• p=“My car has a bad engine.”

• q=“My car has a bad carburetor.”
• pq=“Either my car has a bad engine, or
the downwardmy car has a bad carburetor.” After
pointing “axe” of “”
Meaning is like “and/or” in English.
splits the wood, you
can take 1 piece OR the
other, or both.
69
Disjunction Truth Table
• Note that pq means
p q pq
that p is true, or q is
F
F
true, or both are true!
F T
• So, this operation is
T F
also called inclusive or,
T
T
because it includes the
possibility that both p and q are true.
• “¬” and “” together are also universal.
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Nested Propositional Expressions
• Use parentheses to group sub-expressions:
“I just saw my old friend, and either he’s
grown or I’ve shrunk.” = f  (g  s)
– (f  g)  s would mean something different
– f  g  s would be ambiguous
• By convention, “¬” takes precedence over
both “” and “”.
– ¬s  f means (¬s)  f , not ¬ (s  f)
71
A Simple Exercise
Let p=“It rained last night”,
q=“The sprinklers came on last night,”
r=“The lawn was wet this morning.”
Translate each of the following into English:
¬p
= “It didn’t rain last night.”
“The lawn was wet this morning, and
r  ¬p
= it didn’t rain last night.”
¬ r  p  q = “Either the lawn wasn’t wet this
morning, or it rained last night, or
the sprinklers came on last night.”
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The Exclusive Or Operator
The binary exclusive-or operator “” (XOR)
combines two propositions to form their
logical “exclusive or” (exjunction?).
p = “I will earn an A in this course,”
q = “I will drop this course,”
p  q = “I will either earn an A for this
course, or I will drop it (but not both!)”
73
Exclusive-Or Truth Table
• Note that pq means
p q pq
that p is true, or q is
F
F
true, but not both!
F T
• This operation is
T F
called exclusive or,
T
T
because it excludes the
possibility that both p and q are true.
• “¬” and “” together are not universal.
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Natural Language is Ambiguous
Note that English “or” can be ambiguous
regarding the “both” case! p q p "or" q
“Pat is a singer or
F F
Pat is a writer.” - 
F T
“Pat is a man or
T F
Pat is a woman.” - 
T T
Need context to disambiguate the meaning!
For this class, assume “or” means inclusive.
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The Implication Operator
antecedent
consequent
The implication p  q states that p implies q.
I.e., If p is true, then q is true; but if p is not
true, then q could be either true or false.
E.g., let p = “You study hard.”
q = “You will get a good grade.”
p  q = “If you study hard, then you will get
a good grade.” (else, it could go either way)
76
Examples of Implications
• “If this lecture ends, then the sun will rise
tomorrow.” True or False?
• “If Tuesday is a day of the week, then I am
a penguin.” True or False?
• “If 1+1=6, then Bush is president.”
True or False?
• “If the moon is made of green cheese, then I
am richer than Bill Gates.” True or False?
77
English Phrases Meaning p  q
•
•
•
•
•
•
•
•
“p implies q”
“if p, then q”
“if p, q”
“when p, q”
“whenever p, q”
“p only if q” “
p is sufficient for q”
“q if p”
•
•
•
•
•
“q when p”
“q whenever p”
“q is necessary for p”
“q follows from p”
“q is implied by p”
We will see some equivalent
logic expressions later.
78
The biconditional operator
The biconditional p  q states that p is true if and
only if (IFF) q is true.
p = “Bush wins the 2004 election.”
q = “Bush will be president for all of 2005.”
p  q = “If, and only if, Bush wins the 2004
election, Bush will be president for all of 2005.”
I’m still
here!
2004
2005
79
Boolean Operations Summary
• We have seen 1 unary operator (out of the 4
possible) and 5 binary operators (out of the
16 possible). Their truth tables are below.
p
F
F
T
T
q
F
T
F
T
p pq pq pq pq pq
T F
F
F
T
T
T F
T
T
T
F
F F
T
T
F
F
F T
T
F
T
T
80
Some Alternative Notations
Name:
Propositional logic:
Boolean algebra:
C/C++/Java (wordwise):
C/C++/Java (bitwise):
not and or
  
p pq +
! && ||
~ & |
xor implies



!=
^
iff

==
Logic gates:
81
End of §1.1
You have learned about:
• Propositions: What
they are.
• Propositional logic
operators’
–
–
–
–
Symbolic notations.
English equivalents.
Logical meaning.
Truth tables.
• Atomic vs. compound
propositions.
• Alternative notations.
• Bits and bit-strings.
• Next section: §1.2
– Propositional
equivalences.
– How to prove them.
82
Propositional Equivalence (§1.2)
Two syntactically (i.e., textually) different
compound propositions may be the
semantically identical (i.e., have the same
meaning). We call them equivalent. Learn:
• Various equivalence rules or laws.
• How to prove equivalences using symbolic
derivations.
83
Tautologies and Contradictions
A tautology is a compound proposition that is
true no matter what the truth values of its
atomic propositions are!
Ex. p  p [What is its truth table?]
A contradiction is a compound proposition
that is false no matter what! Ex. p  p
[Truth table?]
Other compound props. are contingencies.
84
Predicate Logic (§1.3)
• Predicate logic is an extension of
propositional logic that permits concisely
reasoning about whole classes of entities.
• Propositional logic (recall) treats simple
propositions (sentences) as atomic entities.
• In contrast, predicate logic distinguishes the
subject of a sentence from its predicate.
– Remember these English grammar terms?
85
Applications of Predicate Logic
It is the formal notation for writing perfectly
clear, concise, and unambiguous
mathematical definitions, axioms, and
theorems (more on these in chapter 3) for
any branch of mathematics.
Predicate logic with function symbols, the “=” operator, and a
few proof-building rules is sufficient for defining any
conceivable mathematical system, and for proving
anything that can be proved within that system!
86
Other Applications
• Predicate logic is the foundation of the
field of mathematical logic, which
culminated in Gödel’s incompleteness
theorem, which revealed the ultimate
limits of mathematical thought:
– Given any finitely describable, consistent
proof procedure, there will still be some
true statements that can never be proven
by that procedure.
Kurt Gödel
1906-1978
• I.e., we can’t discover all mathematical truths,
unless we sometimes resort to making guesses.
87
Subjects and Predicates
• In the sentence “The dog is sleeping”:
– The phrase “the dog” denotes the subject the object or entity that the sentence is about.
– The phrase “is sleeping” denotes the predicatea property that is true of the subject.
• In predicate logic, a predicate is modeled as
a function P(·) from objects to propositions.
– P(x) = “x is sleeping” (where x is any object).
88
Review: Propositional Logic
(§1.1-1.2)
•
•
•
•
•
Atomic propositions: p, q, r, …
Boolean operators:      
Compound propositions: s : (p  q)  r
Equivalences: pq  (p  q)
Proving equivalences using:
– Truth tables.
– Symbolic derivations. p  q  r …
89
Predicates and Quantifiers
variable
predicate
A( x) : Computer x is under attack by an intruder
B( x) : Computer x is functionin g properly
Propositional functions
P( x) : x  3
Q ( x, y ) : x  y  3
R ( x, y , z ) : x  y  z
90
Predicate logic
Computers  {CS1, CS 2, MATH 1}
A( x) : Computer x is under attack by an intruder
A(CS1)  T
A(CS 2)  F
A( MATH 1)  T
B( x) : Computer x is functionin g properly
B (CS1)  F
B (CS 2)  T
B ( MATH 1)  F
91
Universal quantifier:
P( x) : x 1  x
x P(x)
for all x it holds P (x )
(for every element in domain)
x P(x) is true for every real number x
Q( x ) : x 2  0
(for every element in domain)
x Q(x) is not true for every real number x
Counterexample: Q(0)  F
92
Existential quantifier:
x P (x )
there is x such that P (x )
P( x) : x  3
x P (x ) is true because P(4)  T
Q( x) : x  1  1  x  0
x Q (x) is not true
93
For finite domain {x1 , x2 ,, xn }
x P( x)  P( x1 )  P( x2 )    P( xn )
x P( x)  P( x1 )  P( x2 )    P( xn )
94
Quantifiers with restricted domain
x  0 ( x  0)
2
y  0 ( y  0)
3
z  0 ( z 2  2)
Precedence of operators
     

higher
lower
95
Logical equivalences with quantifiers
x( P( x)  Q( x))  xP( x)  xQ( x)
x( P( x)  Q( x))  xP( x)  xQ( x)
x( P( x)  Q( x))  xP( x)  xQ( x) ?
x( P( x)  Q( x))  xP( x)  xQ( x) ?
False
False
96
De Morgan’s Laws for Quantifiers
xP( x)  xP( x)
xP( x)  xP( x)
97
Example
x( P( x)  Q( x))  x( P( x)  Q( x))
 x( P( x)  Q( x))
Recall that:
( p  q)  p  q
98
Translating English into Logical Expressions
P( x)  x is a hummingbir d
Q( x)  x is large bird
R( x)  x lives on honey
S ( x)  x is richly colored
“All hummingbirds are richly colored” x( P( x)  S ( x))
“No large birds live on honey”
x(Q( x)  R( x))
“Birds that do not live on honey
x(R( x)  S ( x))
are dull in color”
“Hummingbirds are small”
x( P( x)  Q( x))
99
Universal Modus Ponens
x( P( x)  Q( x))
P(a), for some particular a in domain
 Q(a)
P (x )
Q (x )
For all positive integers x ,
if x  4
then x 2  2 x
x( P( x)  Q( x))
100  4
P(100)
Therefore, 100  2
2
100
Q(100)
100