Chapter 1: The Foundations: Logic and Proofs
Trần Hòa Phú
Ngày 4 tháng 9 năm 2023
Objectives
Explain what makes up a correct mathematical arguments
Introduce tools to construct correct arguments
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Propositions
(Mệnh đề)
Definition
A proposition is a declarative sentence (that is, a
sentence that declares a fact) that is either true or
false, but not both.
Example
Sun rises in the east and sets in the west.
Hue is the captial of Vietnam.
2+3=5.
2+3=6.
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Which of the following sentences are propositions?
a) 5+7 = 12.
b) x+2 = 11.
c) Answer this question!
d) Today is thursday.
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Chapter 1: The Foundations: Logic and Proofs
Operations of Propositions
Let p, q are propositions.
¬p
p∧q
p∨q
L
p q
p→q
p↔q
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Chapter 1: The Foundations: Logic and Proofs
Proposition ¬p
Definition
Let p be a proposition.
¬p, (p) is the negation of p.
Example
p : My PC runs Windows.
¬p My PC doesn’t run Windows.
Example
p : This book has at least 300 pages.
¬p This book doesn’t have at least 300 pages.
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Find the negation of each of these propositions.
a) Tuan has an iphone
b) There is no pollution in Singapore
c) 2+1 = 3
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Chapter 1: The Foundations: Logic and Proofs
Table 1: The truth table for the negation of a Proposition
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Chapter 1: The Foundations: Logic and Proofs
Proposition p ∧ q
Definition
Let p and q be propositions.
p ∧ q is the proposition "p and q".
Example 1
p : Sun rises in the east
q : Sun sets in the west
p ∧ q : Sun rises in the east and sets in the west.
Example 2
p : Sun is sunning
q : Sun is raining
p ∧ q: "Sun is sunning and it is raining"
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Chapter 1: The Foundations: Logic and Proofs
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Proposition p ∨ q
Definition
Let p and q be propositions.
p ∨ q is the proposition "p or q"
Example
p : Sun rises in the east
q : Sun sets in the west
p ∨ q : Sun rises in the east or it sets in the west.
Example
p : This book has more than 200 pages.
q : This book costs 20 dollars.
p ∨ q: This book has more than 200 pages or it costs 20 dollars.
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Chapter 1: The Foundations: Logic and Proofs
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Proposition p → q
Definition
Let p and q be propositions.
p → q is the proposition "if p, then q", "p is sufficient
condition for q","q is necessary condition for p"
Example
p: Maria learns discrete mathematics.
q: Maria will find a good job.
p → q if Maria learns discrete mathematics, then Maria will find a
good job.
p → q For Maria to get a good job, it is sufficient for her to learn
discrete mathematics.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Determine whether each of these conditional
statements is true or false.
a) If 1 + 1 = 2, then 2 + 2 = 5
b) If 1 + 1 = 3, then 2 + 2 = 4
c) If 1 + 1 = 3, then 2 + 2 = 5
d) If monkeys can fly, then 1 + 1 = 3
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Chapter 1: The Foundations: Logic and Proofs
Proposition p ↔ q
Definition
Let p and q be propositions.
The p ↔ q is the proposition "p if and only if q".
Example
p: You can take the flight.
q: You buy a ticket.
p ↔ q: You can take the flight if and only if you buy a ticket.
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Chapter 1: The Foundations: Logic and Proofs
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Chapter 1: The Foundations: Logic and Proofs
Exercise
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Chapter 1: The Foundations: Logic and Proofs
Proposition p q
L
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Chapter 1: The Foundations: Logic and Proofs
Truth Table Summary of Connectives
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Construct a truth table for each of these
compound propositions.
V
a) p ¬p
b) p ¬p
W
c) (p ¬q) → q
W
d) (p q) → (p q)
W
V
e) (p → q) → (q → p)
f) (p ↔ q) ⊕ ¬p
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Chapter 1: The Foundations: Logic and Proofs
Truth table from Compound Propositions
Establishing truth table of proposition given below
(p ∨ ¬q) → (p ∧ q)
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Chapter 1: The Foundations: Logic and Proofs
Logic and Bit Operations
Definition
Computers represent information using bits.
A bit is a symbol with two possible values, namely, 0(zero) and
1(one).
L
Bit operations: x ∧ y (AND), x ∨ y (OR), x y (XOR) (x , y are
two bits.)
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Chapter 1: The Foundations: Logic and Proofs
Bit string and Operations
Definition
A bit string is a sequence of zero or more bits
The length of this string is the number of bits in the string
Example
10101 is a bit string of length 5.
Definition
Operations of bit string: bitwise OR, bitwise AND, bitwise XOR.
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Chapter 1: The Foundations: Logic and Proofs
Example
Find the bitwise OR, bitwise AND and bitwise XOR of the
bit strings 01 1011 0110 and 11 0001 1101.
Solution
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Find the bitwise OR, bitwise AND, and bitwise
XOR of each of these pairs of bit strings.
a) 101 1110, 010 0001
b) 1111 0000, 1010 1010
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Chapter 1: The Foundations: Logic and Proofs
Exercise
What is the value of x after each of these statements is
encountered in a computer program, if x = 1 before the
statement is reached?
a) if x + 2 = 3 then x := x + 1
b) if (x + 1) = 3 OR (2x + 2 = 3) then x := x + 1
c) if 2x + 3 = 5 AND 3x + 4 = 7 then x := x + 1
d) if (x + 1 = 2) XOR (x + 2 = 3) then x := x + 1
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Chapter 1: The Foundations: Logic and Proofs
Propositional Equivalences
Câu sau đây có ý gì?
"Nếu trời mưa, tôi ở nhà và trời không mưa, tôi cũng ở nhà."
Tautology and Contradiction
Logical Equivalences
De Morgan’s Laws
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Chapter 1: The Foundations: Logic and Proofs
Tautology
Definition: A tautology is a compound proposition that is
always true
Example
p ∨ p is always true.
p p p ∨p
T F
T
F T
T
Therefore, p ∨ p is a tautology
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Chapter 1: The Foundations: Logic and Proofs
Contradiction
Definition: A contradiction is a compound proposition that
is always false.
Example
p ∧ p is always false.
p p p ∧p
T F
F
F T
F
Therefore, p ∧ p is a contradiction.
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Chapter 1: The Foundations: Logic and Proofs
Propositional Equivalence
Definition
The compound propositions p and q are called logically
equivalent if p ↔ q is a tautology (p, q are both True or are
both False).
The notation p ≡ q denotes that p and q are logically
equivalent.
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Chapter 1: The Foundations: Logic and Proofs
Example
Show that ¬(p ∧ q) and ¬p ∨ ¬q are logically equivalent.
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Chapter 1: The Foundations: Logic and Proofs
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Chapter 1: The Foundations: Logic and Proofs
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Chapter 1: The Foundations: Logic and Proofs
De Morgan’s Laws
Theorem
Example Find the negation
"He is an actor and he is rich"
Solution
He is not an actor or he isn’t rich".
Example Find the negation
"John has a cellphone or he has a laptop"
Solution
"John doesn’t have a cellphone
andChapter
doesn’t
have a laptop. "
Trần Hòa Phú
1: The Foundations: Logic and Proofs
p → q ≡ ¬p ∨ q
Problem Find the negation
"If it rains, then I stay at home".
Theorem
p → q ≡ ¬p ∨ q.
p →q ≡p∨q ≡p∧q ≡p∧q
Let p: It rains
q: I stay at home
p → q: "If it rains, then I stay at home".
p →q ≡p∧q
The answer is: "It rains and I don’t stay at home"
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Prove that
(p → q) ∧ (p → q) ≡ q
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Tìm mệnh đề phủ định các mệnh đề sau:
1) "Mọi bông hồng đều đẹp"
2) "Có một chiếc xe hơi giá rẻ"
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Chapter 1: The Foundations: Logic and Proofs
Predicates and Quantifiers
Predicates
Example
Let P(x ): ”x > 3”.
P is the propositional function. P(x ) is the value of the
propositional function P at x .
P(2) : 2 > 3,
P(2) is F
P(4) : 4 > 3
P(4) is T
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Chapter 1: The Foundations: Logic and Proofs
Example
Q(x , y ) : ”x = y + 3”.
What are the truth values of the propositions
Q(1, 2) and Q(3, 0)?
Solution
Q(1, 2) : ”1 = 2 + 3”
Q(1, 2) is F
Q(3, 0) : ”3 = 0 + 3”
Q(3, 0) is T
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Let P(x ): "the word x contains the letter a".
What are these truth values?
a) P(orange)
b) P(lemon)
c) P(true)
d) P(false)
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Chapter 1: The Foundations: Logic and Proofs
Quantifiers
Definition
The universal quantification of P(x ) is the statement
∀xP(x ) : "P(x ) for all values of x in the domain"
∀ : "for every", "for all" ("với mỗi", "với mọi", "với tất cả")
Example
P(x ) : ”x + 1 > x ”, domain consists of all real numbers.
∀xP(x ) ≡ ∀x (x + 1 > x ) :
"for every real number x , x + 1 > x "
∀xP(x ) is T
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Chapter 1: The Foundations: Logic and Proofs
Definition
The universal quantification of P(x ) is the statement
∀xP(x ): "P(x ) for all values of x in the domain"
Example
P(x ) : ”x reads books everyday", domain for x consists of all
student.
∀xP(x )≡ ∀x (”x read books every day”):
"for every student x ,x reads books everyday " "
∀xP(x ) : "Every student reads book everyday".
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Chapter 1: The Foundations: Logic and Proofs
Example
Let Q(x ) : ”x < 2”.
What is the truth values of the quantificaion ∀xQ(x ), where the
domain consists of all real numbers?
Solution
∀xQ(x ) : "for every real number x , x < 2".
So ∀xQ(x ) is F since 3 > 2.
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Determine the truth value of each of these
statements if the domain consists of all integers.
a) ∀n(n + 1 > n)
b) ∀n(3n ≤ 4n)
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Chapter 1: The Foundations: Logic and Proofs
The Existential Quantifier
Definition
The existential quantification of P(x ) is the proposition
∃xP(x ) "there exists an element x in the domain such that P(x )”
∃ : "exists", "some" ("tồn tại", "một vài")
Example
P(x ) : ”x + 1 > x ”, domain consists of all real numbers.
∃xP(x ) ≡ ∃x (x + 1 > x ) : "There exists a real number x such that
x + 1 > x ."
∃xP(x ) is T.
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Chapter 1: The Foundations: Logic and Proofs
Definition
The existential quantification of P(x ) is the proposition
∃xP(x ) : "there exists an element x in the domain such that P(x )”
Example
P(x ) : ”x reads books everyday", domain for x consists of all
student.
∃xP(x ) : "There exists a student x such that x reads books
everyday "
∃xP(x ) :"There exists a student who reads book everyday".
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Chapter 1: The Foundations: Logic and Proofs
Example
P(x ) : ”x = x + 1”. What is the truth value of the quantification
∃xP(x ), where the domain consists of all real numbers?
Solution
∃xP(x ):"there exists a real number x such that x = x + 1"
∃xP(x ) is F.
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Chapter 1: The Foundations: Logic and Proofs
Example
P(x ) : ”x > 2”.
What is the truth value of ∃xP(x ), where the domain consists of
the integers?
Solution
∃xP(x ) :"there exists an integer x such that x > 2."
∃xP(x ) is T since 3 > 2.
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Determine the truth value of each of these
statements if the domain consists of all integers.
a) ∃n(2n = 3n)
b) ∃n(n = −n)
c) ∃n(3n = n + 5)
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Let P(x ) : ”x = x 2”, domain consists of the
integers. What are these truth values?
P(0) P(1) P(2) P(−1) ∃xP(x ) ∀xP(x ).
Solution
∃xP(x ) : ” there exists an integer x such that x = x 2 "
∃xP(x ) is True since 1 = 12
∀xP(x ) : "For every integer x , x = x 2 "
∀xP(x ) is F
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Let P(x ) :" x spends more than five hours every weekday in class",
where the domain for x consists of all students.
Translate these statements into English
a) ∃xP(x )
b) ∀xP(x )
c) ∃x ¬P(x )
d) ∀x ¬P(x )
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Translate these statements into English , where
C (x ) : "x is a comedian", F (x ) : "x is funny"
and the domain consists of all people.
a) ∀x (C (x ) → F (x ))
b) ∀x (C (x ) ∧ F (x ))
c) ∃x (C (x ) → F (x ))
d) ∃x (C (x ) ∧ F (x ))
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Chapter 1: The Foundations: Logic and Proofs
Negating Quantified Expression
Theorem
∀xP(x ) ≡ ∃x P(x )
Example Tìm phủ định của câu "Mọi người Việt Nam đều thích
phở".
Solution
"Có một người VN không thích phở"
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Chapter 1: The Foundations: Logic and Proofs
Example
Find negation of the statement ∀x (x 2 > x )?
Solution
∀x (x 2 > x ) ≡ ∃x (x 2 > x ) ≡ ∃x (x 2 ≤ x )
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Chapter 1: The Foundations: Logic and Proofs
Theorem
∃xP(x ) ≡ ∀x P(x )
Example Tìm câu phủ định của mệnh đề "Có một con khỉ biết
đếm".
Solution
"Mọi con khỉ đều không biết đếm"
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Chapter 1: The Foundations: Logic and Proofs
Example
What is the negation of the statement
∃x (x 2 = 2)?
Solution
∃x (x 2 = 2) ≡ ∀x (x 2 = 2) ≡ ∀x (x 2 ̸= 2)
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Tìm mệnh đề phủ định các mệnh đề sau:
) "Mọi bông hồng đều đẹp"
2) "Có một con mèo có 3 chân"
3) "Mọi xe hơi đều đắt tiền"
4) "Có một chiếc xe máy có giá hơn 200 triệu"
5) "Mọi người đều biết đi xe đạp và thích uống cà
phê".
6) ∀x (x > 1 ∨ x < 0)
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Chapter 1: The Foundations: Logic and Proofs
Translating from English into Logical
Expression
Example
Biến đổi câu sau thành mệnh đề logic "tất cả học sinh trong lớp này
đã học môn giải tích". Miền xác định là tập hợp tất cả con người
Solution
"tất cả học sinh trong lớp này đã học môn giải tích"
≡ "mỗi học sinh x trong lớp này, x đã học môn giải tích".
≡ "Mỗi con người x sao cho nếu x là học sinh trong lớp này thì x
đã học môn giải tích"
Thay "Mỗi" thành ∀, P(x ) : "x là học sinh trong lớp này".
Q(x ) : ”x đã học môn giải tích"
≡ ∀x (P(x ) → Q(x )), domain: tập hợp tất cả con người.
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Biến đổi những câu sau đây thành mệnh đề logic với
miền xác định là những sinh vật sống.
"Tất cả sư tử đều hung dữ"
"Một số sư tử không thích săn mồi"
"Một số sinh vật hung dữ không thích ăn thịt".
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Chapter 1: The Foundations: Logic and Proofs
Solution "Tất cả sư tử đều hung dữ"
≡ "Mỗi sinh vật sống x , nếu x là sư tử thì x hung dữ"
Đặt A(x ) :"x là sư tử", B(x ) : ”x hung dữ"
≡ ∀x (A(x ) → B(x )), domain: creatures.
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Chapter 1: The Foundations: Logic and Proofs
"Một số sư tử không thích săn mồi"
≡ "Một số sinh vật sống x , x là sư tử và x không thích săn mồi"
Đặt A(x ) : ”x là sư tử", B(x ) : ”x thích săn mồi"
≡ ∃x (A(x ) ∧ ¬B(x ))
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Chapter 1: The Foundations: Logic and Proofs
"Một số sinh vật hung dữ không thích ăn thịt"
≡ "Một số sinh vật x , x hung dữ và x không thích ăn thịt".
Đặt A(x ) : ”x hung dữ", B(x ) : ”x thích ăn thịt"
≡ ∃x (A(x ) ∧ ¬B(x ))
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Chapter 1: The Foundations: Logic and Proofs
Nested Quantifier
Example
Assume that the domain for the variables x and y consists
of all real numbers. What is the meaning of the following
statement
∀x ∀y (x + y = y + x )
Solution
∀x ∀y (x + y = y + x ) : "for every real number x for every
real number y ,
or "x + y = y + x for all real numbers x and y ".
∀x ∀y (x + y = y + x ) is T
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Chapter 1: The Foundations: Logic and Proofs
Example
What is the meaning of the statement ∀x ∃y (x + y = 0),
where the domain for x and y consists of all real numbers?
Solution
”∀x ∃y (x + y = 0)”:for every real number x , there exists a
real number y such that x + y = 0.
∀x ∃y (x + y = 0) is T
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Let T (x , y ) : "the student x takes the class y"
where x represents a student in a university and y represent
represent a class.
Translate the logical expression into a sentence
∀y ∃xT (x , y )
a) "For each class there is at least a student taking it"
b) "There is a student taking all classes"
c) Every student is taking at least one class
d) There is a class that every student taking it.
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Suppose L(x , y ) : "x buys Chirstmas gifts for y "
where x and y are members of a family.
Translate the statement into logical expression
"Each family member buys Christmas gifts for any other member
but not for himself or herself"
i) ∃x (¬L(x , x ))
ii) ∀x (¬L(x , x ))
iii) ∀x (∀yL(x , y ))
iv) ∀x (∀y ((y ̸= x ) ⇔ L(x , y )))
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Let P(x): "x is an even number"
where the domain is the set of all integers.
Which propositions are true?


i) ∀x P(x ) → P(x + 1)




ii) ∀x P(x ) ↔ P(x 2)


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Chapter 1: The Foundations: Logic and Proofs
Exercise
Find the truth values of the propositions, where
x,y represent real numbers

∀x ∀y (x = y 2) ∧ (x < y )




∃x ∃y (x = y 2) ∧ (x < y )


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Chapter 1: The Foundations: Logic and Proofs
Exercise
Find the truth value of each of the following ,
where x represents an integer.
∃x [(x > 0) ∧ (x 3 < 1)]
∀x [(x > 0) ∨ (x 3 < 0)]
∃x [(x > 0) ∨ (x 3 < 1)]
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Chapter 1: The Foundations: Logic and Proofs
Negating Nested Quantifiers
Example
Find ∀x ∃y (xy = 1) so that no negation precedes a
quantifer.
Solution
∀x ∃y (xy = 1)
≡ ∃x ∃y (xy = 1)
≡ ∃x ∀y (xy = 1)
≡ ∃x ∀y (xy ̸= 1)
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Chapter 1: The Foundations: Logic and Proofs
Exercise



P(x ) → Q(x )
1. Find ∀x 



P(x , y ) → Q(x , y )
2. Find ∃x ∀y 
3. Find the negation of the statement
"There exists a student who if he graduates, then he will
find a good job".
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Chapter 1: The Foundations: Logic and Proofs
Rules of Inference
Is the following argument is valid?
"If it rains, then I stay at home"
"It rains"
Therefore, "I stay at home"
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Chapter 1: The Foundations: Logic and Proofs
Is the following argument is valid?
"If it rains, then I stay at home"
"It doesn’t rain"
Therefore, "I don’t stay at home"
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Chapter 1: The Foundations: Logic and Proofs
Argument, Conclusion, Premises, Valid
Definition
An argument in propositional logic is a sequence of propositions.
Final proposition is called the conclusion.
All but the final proposition are called premises.
An argument is valid if the truth of all its premises implies that
the conclusion is true.
Example
"If it rains, then I stay at home"
"It rains"
Therefore, "I stay at home"
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Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Modus ponens
Theorem
The following argument is valid
p→q
p
∴q
Example The following argument is valid.
"If you work hard, you pass MAD"
"you work hard"
Therefore, "you pass MAD".
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Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Modus tollens
Theorem
The following argument is valid
p→q
¬q
∴ ¬p
Example The following argument is valid.
"If you are a singer, then you sing well"
"You don’t sing well"
Therefore, "you are not a singer".
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Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Hypothetical syllogism
Theorem
The following argument is valid.
p→q
q→r
∴p→r
Example
"If it rains today, then we will not have dinner outdoor today"
"If we don’t have dinner outdoor today, we will have dinner at home
today "
"If it rains today, we will have dinner at home today"
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Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Disjunctive syllogism
Theorem
The following argument is valid.
p∨q
¬p
∴q
Example 1
"Today is Tuesday or Wednesday"
"Today is not Tuesday"
Therefore "Today is Wednesday"
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Addition
Theorem
The following argument is valid.
p
∴p∨q
Example
"He is tall"
Therefore, he is tall or he can sing well.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Simplification
Theorem
The following argument is valid.
p∧q
∴p
Example The following argument is valid.
He sings well and can play guitars.
Therefore he sings well.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Conjunction
Theorem
The following argument is valid.
p
q
∴p∧q
Example The following argument is valid
She can speak English
She can speak French
Therefore, she can speak English and French.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Valid Argument: Resolution
Theorem
The following argument is valid.
p∨q
¬p ∨ r
∴q∨r
Example
"it is not snowing or John is skiing"
"It is snowing or Jack is playing football"
Therefore, "John is skiing or Jacking is playing football"
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Chapter 1: The Foundations: Logic and Proofs
Theorem
The following argument is NOT VALID
p→q
¬p
∴ ¬q
Example The following argument is NOT VALID.
"Nếu trời mưa thì tôi nghỉ học"
"Trời không mưa"
Vậy tôi không nghỉ học
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Chapter 1: The Foundations: Logic and Proofs
Summary
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Chapter 1: The Foundations: Logic and Proofs
Summary
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Chapter 1: The Foundations: Logic and Proofs
Exercise
What are conclusions can be drawn if premises are
given as below?
"If I win the lottery, then I buy a house"
"If I buy a house, then I have a wife"
"I am single"
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
What are conclusions can be drawn if premises are
given as below?
"If I eat spicy foods, then I have strange dreams"
"I have strange dreams if there is thunder while I
sleep"
"I did not have strange dream."
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Chapter 1: The Foundations: Logic and Proofs
Exercise
What are conclusions can be drawn if the premises
are given as below?
"I am either clever or lucky"
"I am not lucky"
"If I am lucky, then I will win the lottery"
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
What are conclusions can be drawn if premises are given:
"I can speak English or Germany"
"If I can speak Germany, I will study a master degree in
German"
"I do not study a master degree in Germany" ?
A. I can speak both English and Germany
B. I can speak Germany but cannot speak English
C. No conclusion is drawn
D. I can speak English but cannot speak Germany.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Rules of Inference for Quantified
Statements
Theorem
The following argument is valid
∀xP(x ), x ∈ domain D
∴ P(c), c is an element of domain D.
Example 1 The following argument is valid
"All Vietnamese people like Phở "
Therefore, Tuấn like Phở.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Combining Rules of Inference for Propositions and
Quantified Statements
Theorem
The following argument is valid
∀x (P(x ) → Q(x ))
P(a), a is a particular element in the domain
∴ Q(a)
Example The following argument is valid.
"Everyone in this class has taken Calculus"
"Tuan is in this class"
Therefore, Tuan has taken Calculus
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Chapter 1: The Foundations: Logic and Proofs
Example Show that the following argument is valid.
"Everyone in this class has taken MAD"
"Tuan is in this class"
Therefore, Tuan has taken MAD.
Solution
P(x ) : "x is in this class"
Q(x ) : "x has taken MAD"
"Everyone in this class has taken MAD": ∀x (P(x ) → Q(x ))
"Tuan is in this class":
∴ "Tuan has taken MAD" Q(Tuan)
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Chapter 1: The Foundations: Logic and Proofs
Theorem
The following argument is valid
∀x (P(x ) → Q(x ))
¬Q(a), a is a particular element in the domain
∴ ¬P(a)
Example The following argument is valid.
"Everyone in this class has taken English"
"Tuan has not taken English"
Therefore, Tuan is not in this class
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Chapter 1: The Foundations: Logic and Proofs
Example
Show that the following argument is valid.
"Everyone in this class has taken MAD"
"Tuan has not taken MAD"
Therefore, Tuan is not in this class .
Solution
P(x ) : "x is in this class"
Q(x ) : "x has taken MAD"
"Everyone in this class has taken MAD":
∀x (P(x ) → Q(x ))
"Tuan has not taken MAD":¬Q(Tuan)
∴ "Tuan is not in this class"
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Chapter 1: The Foundations: Logic and Proofs
Exercise
What are conclusions can be drawn if the
following premises are given as below?
"Every computer science major has a personal
computer "
"Ralph does not have a personal computer"
"Ann has a personal computer"
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Determine whether each of the following arguments is valid or not
valid.
a) All parrots like fruit. My pet bird is not a parrot. Therefore, my
pet bird does not like fruit.
b) Everyone who eats granola everyday is healthy. Linda is not
healthy. Therefore, Linda does not eat granola every day.
c) If Mai knows French, Mai is smart. But Mai doesn’t know
French. So she is not smart.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Given the premises:
"Every Computer Science major studies Discrete
Mathematics"
"Every Math major studies Calculus"
"An studies Discrete Mathematics"
"Binh studies Calculus"
Which conclusions can be drawn?
i) An is a Computer Science major
ii) Binh is a Math major.
A. Only i B. Only ii. C. Both D. None of them
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Given the premises:
"Lisa is a mathematics and computer science major"
"Every computer science major has a computer"
"Some mathematics majors speak French very well"
Which conclusion can be implied?
A. Lisa has a computer
B. Lisa has a computer and speaks English very well
C. Lisa cannot speak French
D. Lisa has not a computer but speaks French very well
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Chapter 1: The Foundations: Logic and Proofs
Exercise
State true or false
i) 1 + 1 = 3 if and only if 2 + 2 = 3
ii) If it is hot, then it is hot
A. T,F
B. T,T
C. F,T
D. F,F
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Chapter 1: The Foundations: Logic and Proofs
Exercise
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Chapter 1: The Foundations: Logic and Proofs
Exercise
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Find the negation of the statement:
For every student, if this student is a fresh man, then he/she is
taking a math course
A. For every student, if this student is a fresh man, then he/she is
not taking any math course
B. There are some freshman who are not taking any math course
C. For every student, if this student is NOT a fresh man, then
he/she is taking a math course
D. For every student, if this student is NOT a freshman, then
he/she is NOT taking any math course.
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Select correct statement(s). p denotes any proposition.
A. p ∧ T ≡ T
B. p ∨ T ≡ p
C. p ≡ p ∧ p
D. p ∧ ¬p ≡ T
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Let Q(w): "the word w contains the letter e".
Find the truth value of each of following propositions
i) Q(lemon)
ii) Q(false)
A. F,F
B. F,T
C. T,T
D. T,F
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Suppose the variable x represents students,
F(x): "x is a freshman", M(x): "x is math major"
Translate ¬(∀x (¬F (x ) ∨ ¬M(x ))) into English statement.
A. No math major is a freshman
B. Every math major is a freshman
C. Some freshman are math majors
D. None of these
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Which statements are true
i) p ∨ (q ∧ r ) ≡ (p ∧ q) ∨ (p ∧ r )
ii) p → (¬q ∧ r ) ≡ ¬p ∨ ¬(r → q)
A. only i
B. only ii
C. Both
D. Neither
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Suppose the variable x represents students,
F(x):"x is a freshman" M(x): "x is math major"
Translate ∀x (F (x ) → ¬M(x )) into English statement
A. No math major is a freshman
B. Some freshman are math majors
C. Every math major is a freshman
D. None of these
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Write the statement in the "If-then" form
"It is necessary to score a goal to win the match"
Which one is true?
i) If the team scores a goal, they will win the match
ii) If the team doesn’t score a goal, they don’t win the
match
A. Only i
B. only ii
C. Both
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D. Neither
Chapter 1: The Foundations: Logic and Proofs
Exercise
Find the negation of the statement
"If Lan knows Python, then Lan knows calculus"
A. None of these
B. If Lan doesn’t know Python, then Lan doesn’t know
calculus
C. Lan knows calculus and Lan doesn’t know Python
D. If Lan knows Python, then Lan doesn’t know calculus
E. Lan knows Python but Lan doesn’t know calculus
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Which arguments are valid?
i) All Wowy’s fans can sing the song "Thiên đàng". I cannot
sing "Thiên đàng". Therefore, I am not a fan of Wowy.
ii) All Son Tung MTP’s fans like the MV "Chạy ngay đi". I
like this MV. Therefore, I am a fan of Sơn Tùng.
A. only i
B. only ii
C. Neither
Trần Hòa Phú
D. Both
Chapter 1: The Foundations: Logic and Proofs
Exercise
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Chapter 1: The Foundations: Logic and Proofs
Exercise
How many rows are there in the truth table for the
compount proposition
¬(r → ¬q) ∨ (p ∧ ¬r )?
A. None of these
B. 6
C. 4
D. 16
E. 8
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Which propositions are tautology?
i) (p → q) ∧ (¬p → q)
ii) ((p → ¬q) ∧ q) → ¬p
A.Only i B. only ii C. Both
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Neither
Chapter 1: The Foundations: Logic and Proofs
Exercise
Suppose the variable x represents students
F(x)" "x is a freshman", M(x):"x is a math major"
Translate ∀x (M(x ) → F (x )) into English statement
A. Some freshman are math majors
B. Every math major is a freshman
C. No math major is a freshman
D. None of these
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Suppose the truth value of p → ¬(p ∧ ¬q) is False
Find the truth value of p, q.
A. F,F
B. T,F
C. F,T
D. T,T
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Suppose the variable x represents students,
F(x): "x is a freshman" M(x):"x is a math major"
Translate ∀x (M(x ) → ¬F (x )) into English statement
A. Every math major is a freshman
B. Some freshman are math majors
C. No freshman is a math major
D. None of these
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs
Exercise
Using c for "it is cold" and r for "it is rainy"
Write " It is rainy if it is not cold" by symbols.
i) ¬c → r
ii) r → ¬c
iii) c → r
iv) ¬c ∧ r
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Chapter 1: The Foundations: Logic and Proofs
Exercise
Let T (x , y ) : "Student x has studied subject y"
R(x , z): "student x is in class z".
Translate the following sentence into a logical expression
"Every student in class MAD101 has studied Calculus".
i) ∀x (T (x , Calculus) → R(x , MAD101))
ii) ∀x (R(x , MAD101) → T (x , MAD101))
iii) ∃x (T (x , Calculus) → R(x , MAD101))
iv) ∃x (R(x , MAD101) → T (x , Calculus))
Trần Hòa Phú
Chapter 1: The Foundations: Logic and Proofs