# Document ```CS100 : Discrete Structures
Chapter 1
Logic Theory
Department of Computer Science
Office # SR 068
Tel # 2581888
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Propositional Logic
 Proposition: A proposition is a declarative statement
( a statement that declares a fact) that is either
TRUE or FALSE, but not both.
 The area of logic that deals with propositions is
called propositional logic.
Propositions
Not Propositions
Riyadh is the capital of Saudi Arabia
How many students in this class?
Every cow has 4 legs.
Bring me coffee!
3 + 2 = 32
4+3=7
X+2=3
Y+Z=X
&quot;x +2=3&quot;, where x is a variable representing a number, is not a proposition ,
because unless a specific value is given to x we can not say whether it is true or false, nor do we
know what x represents .
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Propositional Logic - Negation
We use letters to denote propositional variables
Truth value of true proposition is denoted by T
Truth value of false proposition is denoted by F
Negation: Suppose p is a proposition. The negation of
p is written p and has meaning: “It is not the case
that p.”
 The proposition p is read “NOT p”
 Example: p: “Today is Friday”
p: “Today is NOT Friday”
 Truth table for negation:
p
p




Notice that p is a
proposition!
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T
F
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F
T
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Propositional Logic - Conjunction
 Conjunction corresponds to English “and.”
 Conjunction: Let p and q be two propositions. The
conjunction of p and q, denoted by p  q, is the
proposition “p and q”. The p  q is true when both p
and q are true, otherwise false.
 Example: p: “Today is Friday”
q: “It is raining today”
p  q: “Today is Friday and it is raining today”
 Truth table for conjunction:
P
q
pq
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T
T
F
F
T
F
T
F
T
F
F
F
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Propositional Logic - Disjunction
 Disjunction corresponds to English “or.”
 Disjunction: Let p and q be two propositions. The
disjunction of p and q, denoted by p  q, is the
proposition “p or q”. The p  q is false when both p
and q are false, otherwise true.
 Example: p: “Today is Friday”
q: “It is raining today”
p  q: “Today is Friday or it is raining today”
 Truth table for disjunction:
p
q
pq
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T
T
F
F
T
F
T
F
T
T
T
F
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Propositional Logic – Exclusive Or
 Exclusive Or: Let p and q be two propositions. The
exclusive or of p and q, denoted by p ⊕ q, is the
proposition that is true when exactly one of p and q
is true, otherwise false.
 Example: p: “Today is Friday”
q: “It is raining today”
p ⊕ q: “Today is Friday exclusive or it is raining
today”
p
q
p⊕q
 Truth table for Exclusive Or:
T
T
F
F
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T
F
T
F
F
T
T
F
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Propositional Logic – Conditional
Statement
 It corresponds to English “if p then q” or “p implies q.”
 Conditional: Let p and q be two propositions. The
conditional statement (implication) p  q is the
proposition “if p, then q”. The conditional statement
p  q is false when p is true and q is false, otherwise
true. (p is called the hypothesis ,q is the conclusion)
 Example: If it is raining, then it is cloudy.
 If I am elected, then I will lower taxes.
 If you get 100% in the final, then you will get
p
q
pq
 Truth table for implication:
T
T
T
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T
F
F
F
T
F
F
T
T
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Propositional Logic - Special Definitions
 Converse: q  p is converse of p  q.
 Ex. p  q: “If it is noon, then I am hungry.”
q  p: “If I am hungry, then it is noon.”
 Contrapositive: q  p is contapositive of p  q.
 Ex. p  q: “If it is noon, then I am hungry.”
q  p: “If I am not hungry, then it is not noon.”
 Inverse: p  q is inverse of p  q.
 Ex. p  q: “If it is noon, then I am hungry.”
p  q: “If it is not noon, then I am not hungry.”
p  q has same truth values as q  p
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Propositional Logic - Questions
 Q1: Give some examples of p  q, then tell what is
q  p
 q  p
 p  q
 Q2: State the converse, contrapositive, and inverse
of each of them
 If it rains today, then I will stay at home
 I come to class whenever there is going to be a
examination
 A positive integer is prime only if it has no
divisors other than 1 and itself.
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Propositional Logic – Biconditional
Statement
 It corresponds to English “p if and only if q”.
 Biconditional: Let p and q be two propositions. The
biconditional statement (bi-implication) p ↔ q is
the proposition “if p and only if q”. The
biconditional statement p ↔ q is true when p and
q have the same truth values, otherwise false.
 p ↔ q has same truth value as (p q)  (q  p).
 Example: p: “You can take the flight”
p
q
p↔q
q : “You buy a ticket”
T
T
T
p ↔ q : “You can take the
T
F
F
flight if and only if
F
T
F
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F
F
T
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Propositional Logic – Precedence
 Precedence of Logical Operators:
 P Λ &not; q νr →s ↔w νm
 (((P Λ (&not; q)) ν r) →s) ↔(w νm)
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Operator
Precedence
&not;
Λ
ν
→
↔
1
2
3
4
5
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Propositional Logic – Precedence
 A compound proposition that is:
(1) always true is called a tautology
(2) always false is called a contradiction
(3) neither a tautology nor a contradiction is called
contingency or satisfiable.
p
p
p  p
p  p
T
F
T
F
F
T
T
F
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Propositional Logic – Compound
Propositions
 Construct truth table for (p  q) → (p  q).
p
q
q p  q p  q (p  q) → (p  q)
T
T
F
F
T
F
T
F
F
T
F
T
T
T
F
T
T
F
F
F
T
F
T
F
 Q3: Construct truth table for
 (p  q) ↔ (p  q).
 (p ⊕ q) → (p ⊕ q).
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Propositional Logic – Translating English
Sentences
 The sentence “The automated reply cannot be sent
when the file system is full” can be translated as
q → p
where p: “The automated reply can be sent ”
q: “The file system is full”
 The sentence “You cannot drive a car if you are
under 4 feet tall unless you are older than 16 years
old” can be translated as
(q r) → p
where p: “You can drive a car ”
q: “You are under 4 feet tall ”
r: “You are older than 16 years old”
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Propositional Logic – Translating English
Sentences
 The sentence “You can access the Internet from
campus only if you are a computer science major or
you are not a freshman” can be translated as
p → (q  r)
where p: “You can access the Internet from campus”
q: “You are a computer science major”
r: “You are a freshman”
 Q4: Translate the following sentences into logical
expressions:
 “Access is granted whenever the user has paid
the subscription fee and enters a valid password”
 “If the user has not entered a valid password but
has paid the subscription fee, then access is
granted”
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Propositional Equivalences – Logic and
Bit Operations
 Bit: A bit is a symbol with two possible values,
namely, 0 (zero) and 1 (one).
 A bit can be used to represent a truth value as 1 for
T and 0 for F
 Bit string: A bit string is a sequence of bits. The
length of the string is number of bits in the string.
 Example: 10101001 is a bit string of length eight
 We define the bitwise OR, AND, and XOR of two
strings of same length to be the strings that have as
their bits the OR, AND, and XOR of the
corresponding bits in the two strings, respectively.
 We use the symbols , , and ⊕ to represent bitwise
OR, AND, and XOR, respectively.
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Propositional Equivalences – Logic and
Bit Operations
 Truth table for bitwise OR, AND, and XOR:
x
y
xy
xy
x⊕y
0
0
1
1
0
1
0
1
0
1
1
1
0
0
0
1
0
1
1
0
 Q5: Find bitwise OR, AND, and XOR of the bit strings
 01 0011 0100 and 10 1001 0101
 00 1011 1010 and 11 1010 0101
 11 0101 0110 and 10 1101 0010
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Propositional Logic - Applications
 We are using propositional logic as a foundation for
formal proofs.
 Propositional logic is also the key to writing good
code…you can’t do any kind of conditional (if)
statement without understanding the condition
you’re testing.
 All the logical connectives we have discussed are also
found in hardware and are called “gates.”
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Propositional Equivalences – Logical
Equivalences
 Equivalence: The compound propositions p and q are
logically equivalent if p↔q is a tautology. In other
words, p and q are logically equivalent if their truth
tables are the same. We write p  q.
 Example: (p  q)  p  q.
 Truth tables for (p  q) and p  q :
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p
q
T
T
F
F
T
F
T
F
p  q (p  q)
T
T
T
F
F
F
F
T
p
q
p  q
F
F
T
T
F
T
F
T
F
F
F
T
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Propositional Equivalences – Logical
Equivalences
 Example: p  q  p  q.
 Truth tables for p  q and p  q:
p
q
pq
p
p  q
T
T
F
F
T
F
T
F
T
F
T
T
F
F
T
T
T
F
T
T
 Q5: Show that p  (q  r)  (p  q)  (p  r)
 Note: if we have n logical variable then the size of
the truth table is 2n
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Propositional Equivalence – Prove the following
Logical Equivalences
Equivalence
Name
p Λ T  p and p ν F  p
Identity laws
p ν T  T and p Λ F  F
Domination laws
p ν p  p and p Λ p  p
Idempotent laws
&not;( &not; p)  p
Double negation law
p ν q  q ν p and p Λ q  q Λ p
Commutative laws
(p ν q) ν r  p ν (q ν r) and (p Λ q) Λ r  p Λ (q Λ
r)
Associative laws
p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p  r)
Distributive laws
&not;(p ν q)  &not;p Λ &not;q and &not;(p Λ q)  &not;p ν &not;q
De Morgan’s laws
p  (p  q)  p and p  (p  q)  p
Absorption laws
p ν &not;p  T and p Λ &not;p  F
Negation laws
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Propositional Equivalence – Prove the
following Logical Equivalences
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Propositional Equivalences – Another
way of proving Logical Equivalences
 In addition to truth table we can use logical
identities that we already know to establish new
logical equivalences
 Examples:
 &not;(p  q)  p &not;q (see p.26)
 Show that &not;(p (&not; p  q)) and &not; p  &not; q are equivalent.
 Show that p q  p  q is a tautology .(see p.27
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Predicate Logic
 Introduction:
 x&gt;20 (We must know x to tell if it is true or false)
 All students passed. (We have to write a proposition
for every student asserting that he passed and take
the conjunction of all these propositions)
 Some students didn’t pass (Can you make such an
assertion without referring to a particular student?)
 X is father of y
The above assertions are not propositions. However,
they appear quite often in mathematics and we want
to do deal with such assertions.
Predicate logic is a more powerful logic than
propositional logic and it addresses some of the
limitations of propositional logic
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Predicate Logic
 Predicate: refers to a property that the subject of
the statement can have or relationships among
other subjects
 Examples
 X&gt;20. This statements has two parts.
The variable x and is called the subject
The property “is greater than 20” and is called the
predicate
We can also write P(x) where P is the predicate ‘’is
greater than 20” and x is the subject. P(x) is called
the propositional function P at x
 Note: once a value has been assigned to the variable x
the statement P(x) becomes a proposition that has a
truth value. For example P(8) is the statement that
8&gt;20 which is false
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Predicate Logic
Examples
x is father of y. This statements has two
parts.
The variables x and y and are called the subjects
The relationship “is father of ” and is called the
predicate
We can also write P(x,y) where P is the predicate
‘is father of ” and x,y are the subjects. P(x,y) is
called the propositional function P at x and y.
As we have seen predicate logic can describe
both relationships and properties of subjects
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Predicate Logic: Quantifiers
Quantification expresses the extent to which
a predicate is true over a range of elements.
In English: all,some,none anf few are used.
All birds have wings
Some birds do not fly.
The area of logic that deals with predicates
and quantifiers is called predicate calculus or
first order logic.
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Predicate Logic: Universal Quantifier
 Universal Quantifier: Asserts that a property is true for
all values of a variable in a particular domain.
 Let P(x) be a propostional function.The universal quantifier
of P(x) is the statement:
 P(x) is true for all x in the domain.
 The domain is called the universe of discourse
 We write x P ( x)
 We read it as “for all x P(x)”
 An element for which P(x) is false is called counterexample
of x P ( x)
 The statement is either true of false, sox P ( x)
is a proposition.
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Predicate Logic: Universal Quantifier
 Examples:
 Let P(x) be “x2 &gt; 0” then
 Statement x P ( x) is not true when the universe of
discourse consists of all real numbers
 P(0) is false ( counterexample)
 Statement x P ( x) is true when the universe of discourse
consists of all real numbers which are greater than 0.
 What is the truth value of x( x 2  x)
consists of all real numbers?
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if the domain
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Predicate Logic: Existential Quantifier
 Existential Quantifier: Asserts that there is an element
with a certain property in a particular domain.
 Let P(x) be a propostional function.The Existential quantifier
of P(x) is the statement:
 There exists an element x in the domain such that P(x) is
true.
 The domain is called the universe of discourse
 We write x P ( x )
 We read it as “There is an x such that P(x)”
 or “For some x P(x)”
 The statement x P ( x ) is false iff there is no element x in
the domain for which P(x) is true
 The statement is either true of false, so
is a proposition.
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Predicate Logic: Existential Quantifier
 Examples:
Let P(x) be “x2 &gt; 0” then
 Statement x P ( x) is true since for example P(2) is
true
 Statement x P ( x) is false iff there is no element in
the domain for which P(x) is true.
 Finding at least one element for which P(x) is true is
sufficient to asserts that x P ( x) is true.
 What is the truth value of x ( x  3) if the domain
consists of all real numbers?
 What is the truth value of x( x  1  x) if the domain
consists of all real numbers?
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Predicate Logic: Quantifiers
 When all the elements in domain can be listed as:x1,x2,…,xn then:

is the same as P(x1) P(x2)…  P(xn)
 xP(x ) is the same as P(x1)  P(x2)…  P(xn)
 Quantifiers with restricted domain
x  0( x 2  0) (note that it is the same as x( x  0  x 2  0)
y  0( y 3  0)
z  R( z 2  1)
z  R( z 2  1)
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The two quantifiers have precedence over all
logical operators from propositional logic.
x P( x)  Q( x) means (x P( x))  Q( x)
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Predicate Logic: Negation of
Quantified Expression
 Examples
 Every Students in this class has taken calculus course
 Can be represented as: x P ( x) where P(x) is”x has taken
calculus course”
 To negate it we write “there is a student in this class who has
not taken calculus course” which can be translated to x  P ( x )
 There is a man who can fly
 Can be represented as: x F ( x) where F(x) is”x can fly”
 To negate it we write “There is no man who can fly”
 Which can be stated as ”every man can not fly” x F ( x)
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Predicate Logic: From English to
Logic(1)
 Every student in the class has taken calculus course
 1- For every student x in the class, x has taken calculus
 2x T ( x)
 where T(x) is “x has taken calculus”
 The domain of x is students of this class
 For every Man, if the man has a job then he has a car
 1- For every man x, if x has a job then x has a car
x ( J ( x)  C ( x)) where
 2 J(x) is “x has a job”
 C(x) is “x has a car”
 The domain of x is all the men
 Some student in this class has visited Abha or Dammam
 For every Saudi, if he is a student then he has visitd Abha
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x( A( x)  D( x))
x( S (x)  A( x))
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Predicate Logic: Nested Quantifiers
 (1) The statement xy ( x  y  0) says that
For every number x, there is a number y such that x+y=0 (additive inverse)
 (2) The statement
negative integers)
xy ( y  x ) says that (the domain is all non-
There exists x, such that for every number y, y is greater than or equal
to x (0 is the smallest non-negative number)
 (3) The statement
xy( x  y  y  x) says that
X+y=y+x for all numbers x and y (commutative law of addition).
 (4) The statement
xy( x  y  17 )
says that
There exist two numbers x and y such that their product equals 17
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Predicate Logic: From English to Logic
and vice versa(2)
 The sum of two positive integers is always positive
xy ( ( x  0)  ( y  0)  x  y  0)
 Every real number except zero has a multiplicative inverse
x( x  0  y ( xy  1))

or
x  0 y ( xy  1)
 Translate(p.55) x(C ( x)  y (C ( y )  F ( x, y )))
xy ( xy  y )
 Translate
 If a person is a male and is a parent then this person is
someone’s father. x(Male(x)Parent(x)y Father(x,y))
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Predicate Logic: From English to
Logic(3)
Example: Let
 1) S(x) be “x is s student” 2)F(x) be “x is a faculty member”
 3) E(x,y) be “x has sent an email to y” and the domain is every person in our college.
Translate the following sentences into logical Expressions:
 Every person in the college has sent an email to every person including himself
 xy(E(x,y))
 Every person in the college has sent an email to every other person not including
himself
 xy(xy  E(x,y))
 At least one email has been sent from one person to another
 xy(xyE(x,y))
 There is a person who sent an email to every other person
 xy(xyE(x,y))
 Every person has been sent at least one email
 xy(xyE(y,x))
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Predicate Logic: From English to
Logic(4)
 Example: Let
 1) S(x) be “x is s student” 2)F(x) be “x is a faculty member”
 3) E(x,y) be “x has sent an email to y” and the domain is every person in our college.
Translate the following sentences into logical Expressions:
 Ayman has sent an email to Dr.Saad:
 Every student sent an email to Dr.Zakir
 x(S(x)E(x,Dr.Zakir))
 Every Faculty member has either sent an email to Dr.Ahmad or has been sent an
 Some student has not sent an email to any faculty member
 x(S(x) y(F(y)E(x,y))
 There is a faculty member who has never been sent an email by a student.
 x(F(x) y(S(y)E(y,x))
 Some student has sent an email to every faculty member
 x(S(x) y(F(y)E(x,y))
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Predicate Logic: Negating Nested
Quantifiers
 Negating Nested Quantifiers
 Negate x y( xy  1)




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1- x y( xy  1)
2- x y( xy  1)
3- x y( xy  1)
4- x y( xy  1)