Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Chapter 1
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Logics
Discrete Structures for Computing on January 4, 2023
Contents
Propositional Logic
Logical Equivalences
Exercise
Nguyen An Khuong, Tran Tuan Anh, Nguyen Tien Thinh, Mai
Xuan Toan, Tran Hong Tai
Faculty of Computer Science and Engineering
University of Technology - VNUHCM
trtanh@hcmut.edu.vn
1.1
Contents
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
1
Propositional Logic
Contents
Propositional Logic
2
Logical Equivalences
3
Exercise
Logical Equivalences
Exercise
1.2
Course outcomes
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Course learning outcomes
L.O.1
Understanding of logic and discrete structures
L.O.1.1 Describe definition of propositional and predicate logic
L.O.1.2 Define basic discrete structures: set, mapping, graphs
L.O.2
Represent and model practical problems with discrete structures
L.O.2.1 Logically describe some problems arising in Computing
L.O.2.2 Use proving methods: direct, contrapositive, induction
L.O.2.3 Explain problem modeling using discrete structures
L.O.3
Understanding of basic probability and random variables
L.O.3.1 Define basic probability theory
L.O.3.2 Explain discrete random variables
L.O.4
Compute quantities of discrete structures and probabilities
L.O.4.1 Operate (compute/ optimize) on discrete structures
L.O.4.2 Compute probabilities of various events, conditional
ones, Bayes theorem
Toan, Tran Hong Tai
Contents
Propositional Logic
Logical Equivalences
Exercise
1.3
Logic
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Definition (Averroes)
The tool for distinguishing between the true and the false.
Contents
Definition (Penguin Encyclopedia)
The formal systematic study of the principles of valid inference
Propositional Logic
Logical Equivalences
Exercise
and correct reasoning.
Definition (Discrete Mathematics - Rosen)
Rules of logic are used to distinguish between valid and invalid
mathematical arguments.
1.4
Applications in Computer Science
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
•
Design of computer circuits
•
Construction of computer programs
•
Verification of the correctness of programs
•
Constructing proofs automatically
•
Artificial intelligence
•
Many more...
Contents
Propositional Logic
Logical Equivalences
Exercise
1.5
Propositional Logic
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Definition
A proposition is a declarative sentence that is either true or false,
but not both.
Contents
Propositional Logic
Examples
•
Hanoi is the capital of Vietnam.
•
New York City is the capital of USA.
Logical Equivalences
Exercise
• 1+1=2
• 2+2=3
1.6
Examples
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Examples (Which of these are propositions?)
•
How easy is logic!
•
Read this carefully.
•
H1 building is in Ho Chi Minh City.
• 4>2
• 2n ≥ 100
• The Sun circles the
• Today is Thursday.
•
Contents
Propositional Logic
Logical Equivalences
Exercise
Earth.
Proposition only when the time is specified
1.7
Notations
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Contents
•
Propositions are denoted by
•
The
truth value
p, q, . . .
(ch¥n trà) is true (T) or false (F)
Propositional Logic
Logical Equivalences
Exercise
1.8
Operators
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Negation
- Phõ ành:
¬p
Contents
B£ng: Truth Table for Negation
p
¬p
T
F
F
T
Propositional Logic
Logical Equivalences
Exercise
1.9
Operators
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Conjunction
- Hëi:
p and
p∧q
Disjunction
q
- Tuyºn:
p or
p∨q
q
Contents
p
q
p∧q
p
q
p∨q
T
T
T
T
T
T
T
F
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
I'm teaching DM1 and it is
We need students who have
raining today.
experience in Java or C++.
Propositional Logic
Logical Equivalences
Exercise
Tomorrow, I will eat Pho or Bun
bo.
1.10
Operators
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Exclusive OR
p or
q
- Tuyºn lo¤i :
p⊕q
Implication
(but not both)
if
- K²o theo :
p,
then
p→q
q
Contents
Propositional Logic
p
q
p⊕q
p
q
p→q
T
T
F
T
T
T
T
F
T
T
F
F
F
T
T
F
T
T
F
F
F
F
F
T
Logical Equivalences
Exercise
If it rains, the pavement will be
wet.
1.11
More Expressions for Implication p → q
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
•
then
q
• p
• p
implies
q
• q
• p
if
only if
q
• q
unless
¬p
if
p,
is sufficient for
Contents
q
Propositional Logic
p
Logical Equivalences
Exercise
•
If you get 100% on the final, you will get 10 grade.
•
If you feel asleep this afternoon, then
2 + 3 = 5.
1.12
Conditional Statements From p → q
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
• q → p (converse - £o )
• ¬q → ¬p (contrapositive
•
Contents
- ph£n £o )
Prove that only contrapositive have the same truth table with
Propositional Logic
Logical Equivalences
Exercise
p→q
1.13
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Exercise
What are the converse and contrapositive of the following
conditional statement
If he plays online games too much, his girlfriend leaves him.
•
Converse: If his girlfriend leaves him, then he plays online
Contents
Propositional Logic
Logical Equivalences
Exercise
games too much.
•
Contrapositive: If his girlfriend does not leave him, then he
does not play online games too much.
1.14
Biconditionals
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
p↔q
p if and only if
p
q
p↔q
T
T
T
T
F
F
F
T
F
F
F
T
• p is necessary and sufficient
• if p then q , and conversely.
• p
iff
q
for
Contents
Propositional Logic
Logical Equivalences
Exercise
q .
q .
1.15
The order of operators
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
•
1. in the bracket()
•
2. negation
•
3.
∨, ∧, ⊕
•
4.
→
•
5.
↔
¬
Contents
Propositional Logic
Logical Equivalences
Exercise
1.16
Translating Natural Sentences
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Exercise
I will buy a new phone
4
or
only if
I have enough money to buy iPhone
my phone is not working.
• p:
• q:
I have enough money to buy iPhone 4
• r:
My phone is working
I will buy a new phone
Contents
Propositional Logic
Logical Equivalences
Exercise
• p → (q ∨ ¬r)
1.17
Translating Natural Sentences
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Exercise
Contents
Propositional Logic
He will not run the red light if he sees the police unless he is too
Logical Equivalences
risky.
Exercise
1.18
Construct Truth Table
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Exercise
Construct the truth table of the compound proposition
(p ∨ ¬q) → (p ∧ q).
Contents
Propositional Logic
Logical Equivalences
¬q
p ∨ ¬q
p∧q
(p ∨ ¬q) → (p ∧ q)
T
F
T
T
T
F
T
T
F
F
F
T
F
F
F
T
F
F
T
T
F
F
p
q
T
T
Exercise
1.19
Exercise - Truth table
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
¬p → (¬q ∨ r)
p
q
r
¬p
¬q
¬q ∨ r
¬p → (¬q ∨ r)
T
T
T
F
F
T
T
T
T
F
F
F
F
T
T
F
T
F
T
T
T
T
F
F
F
T
T
T
F
T
T
T
F
T
T
F
T
F
T
F
F
F
Propositional Logic
F
F
T
T
T
T
T
Logical Equivalences
F
F
F
T
T
T
T
Exercise
a)
(p ∧ q) → ¬q
b)
(p ∨ r) → (r ∨ ¬p)
c)
(p → q) ∨ (q → p)
d)
(p ∨ ¬q) ∧ (¬p ∨ q)
e)
(p → ¬q) ∨ (q → ¬p)
f)
¬(¬p ∧ ¬q)
g)
(p ∨ q) → (p ⊕ q)
h)
(p ∧ q) ∨ (r ⊕ q)
Contents
1.20
Applications
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
•
System specifications
•
When a user clicked on
Help button, a pop-up will be shown
up
•
Boolean search
•
•
Contents
Propositional Logic
Logical Equivalences
Exercise
type dai hoc bach khoa in Google
means dai AND hoc AND bach AND khoa
1.21
Applications (cont.)
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
• Logic puzzles
•
There are two kinds of inhabitants on an island, knights, who
always tell the truth, and their opposites, knaves, who may
A and B . What are A and B
B says The two of us are
lie. You encounter two people
A
says B is a knight and
opposite types ?
if
Contents
Propositional Logic
Logical Equivalences
Exercise
• Bit operations
•
101010011 is a bit string of length nine.
1.22
Tautology and Contradiction
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Definition
A compound proposition that is always true (false) is called a
tautology - h¬ng óng (contradiction - h¬ng sai).
•
Tautology: h¬ng óng
•
Contradiction: m¥u thu¨n
Contents
Propositional Logic
Logical Equivalences
Exercise
Example
• p ∨ ¬p
• p ∧ ¬p
(tautology)
(contradiction)
1.23
Question
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Which of the following is a tautology
Hint: Apply truth table.
a)
(p ∨ q) → (p ∧ q)
b)
(p ∧ q) → (p ∨ q)
c)
p → (¬q → p)
d)
p → (p → q)
e)
p → (p → p)
f)
(p → q) → [(p → r) → (q → r)]
Contents
Propositional Logic
Logical Equivalences
Exercise
1.24
Proposition? Truth value?
Logics
Nguyen An Khuong,
a) Fansipan is the highest mountain in Vietnam.
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
b) Two coprime numbers have the only common divisor of 1.
c)
Toan, Tran Hong Tai
The product of 3 continuous integers is divisible by 3.
d) Stand up!
e)
x+1=0
f)
Hexagons have 8 vertices.
g) 0 is a positive number.
h) The equation:
x2 + 5x + 6 = 0
i)
is 2 a prime number?
j)
The equation
Contents
has no root.
mx2 + 2x − 1 = 0
Propositional Logic
has a single root if and only if m=-1.
l)
x
2
Logical Equivalences
Exercise
k) There is a prime that is even.
+ 1 > 0.
m) When will our class go camping?
n) Mercury is not a metal.
20
o) 3
>
230 .
p) Airplanes are the fastest transport.
q) 2002 is a leap year.
r)
There are infinite prime numbers.
s)
10
2
t)
No smoking in public place.
−1
is divisible by 11.
u) All even positive integer is a summation of 2 prime numbers.
v) x is a prime number if it doesn't have any divisor other than 1 and
x.
1.25
Logical Equivalences
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Definition
The compound compositions
if
p↔q
p
q are
p ≡ q.
and
is a tautology, denoted
called logically equivalent
Contents
Propositional Logic
Logical Equivalences
Exercise
Example
Show that
¬(p ∨ q)
and
¬p ∧ ¬q
are logically equivalent.
1.26
Logical Equivalences
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
p∧T
p∨F
≡
≡
p
p
p∨T
p∧F
≡
≡
T
F
Domination laws
p∨p
p∧p
≡
≡
p
p
Idempotent laws
¬(¬p)
≡
p
Double negation law
Identity laws
Luªt çng nh§t
Luªt nuèt
Contents
Propositional Logic
Logical Equivalences
Exercise
Luªt lôy ¯ng
Lu¥t phõ ành k²p
1.27
Logical Equivalences
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
p∨q
p∧q
≡
≡
q∨p
q∧p
(p ∨ q) ∨ r
(p ∧ q) ∧ r
≡
≡
p ∨ (q ∨ r)
p ∧ (q ∧ r)
Associative laws
p ∨ (q ∧ r)
p ∧ (q ∨ r)
≡
≡
(p ∨ q) ∧ (p ∨ r)
(p ∧ q) ∨ (p ∧ r)
Distributive laws
¬(p ∧ q)
¬(p ∨ q)
≡
≡
¬p ∨ ¬q
¬p ∧ ¬q
De Morgan's law
p ∨ (p ∧ q)
p ∧ (p ∨ q)
≡
≡
p
p
Commutative laws
Luªt giao ho¡n
Luªt k¸t hñp
Luªt ph¥n phèi
Contents
Propositional Logic
Logical Equivalences
Exercise
Luªt De Morgan
Absorption laws
Luªt hót thu
1.28
Logical Equivalences
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Equivalence
p ∨ ¬p
p ∧ ¬p
p→q
(p → q) ∧ (p → r)
(p → r) ∧ (q → r)
(p → q) ∨ (p → r)
(p → r) ∨ (q → r)
p↔q
p↔q
≡
≡
≡
≡
≡
≡
≡
≡
≡
T
F
¬p ∨ q
p → (q ∧ r)
(p ∨ q) → r
p → (q ∨ r)
(p ∧ q) → r
(p → q) ∧ (q → p)
(¬p ∨ q) ∧ (p ∨ ¬q)
Contents
Propositional Logic
Logical Equivalences
Exercise
1.29
Constructing New Logical Equivalences
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Example
Show that
¬(p ∨ (¬p ∧ q))
and
¬p ∧ ¬q
are logically equivalent by
developing a series of logical equivalences.
Solution
Contents
¬(p ∨ (¬p ∧ q))
Consequently,
Propositional Logic
≡
¬p ∧ ¬(¬p ∧ q)
by the second De Morgan law
≡
¬p ∧ [¬(¬p) ∨ ¬q]
by the first De Morgan law
≡
¬p ∧ (p ∨ ¬q)
by the double negation law
≡
(¬p ∧ p) ∨ (¬p ∧ ¬q)
by the second distributive law
≡
F ∨ (¬p ∧ ¬q)
because
≡
¬p ∧ ¬q
by the identity law for
¬(p ∨ (¬p ∧ q))
and
¬p ∧ ¬q
Logical Equivalences
Exercise
¬p ∧ p ≡ F
F
are logically equivalent.
1.30
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Negate the following proposition and try to simplify it.
Example
p → (¬q ∧ r)
By using the truth table, we can prove that
p→q
logical equivalence.
¬(p → (¬q ∧ r))
≡ ¬(¬p ∨ (¬q ∧ r))
≡ p ∧ ¬(¬q ∧ r)
≡ p ∧ (q ∨ ¬r)
Negate:
a)
p ∧ (q ∨ r) ∧ (¬p ∨ ¬q ∨ r)
b)
(p ∧ q) → r
c)
p ∨ q ∨ (¬p ∧ ¬q ∧ r)
d)
[[[(p ∧ q) ∧ r] ∨ [(p ∧ r) ∧ ¬r]] ∨ ¬q] → s
and
¬p ∨ q
are
Contents
Propositional Logic
Logical Equivalences
Exercise
1.31
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Prove the following proposition are logical equivalence.
Hint: Apply truth table or the series of logical equivalences.
Contents
a)
¬(p ↔ q)
b)
(p → q) ∧ (p → r)
v
p → (q ∧ r)
c)
(p → r) ∧ (q → r)
v
(p ∨ q) → r
d)
(p → q) ∨ (p → r)
v
p → (q ∨ r)
e)
¬p → (q → r)
f)
p↔q
v
v
¬p ↔ q
v
Propositional Logic
Logical Equivalences
Exercise
q → (p ∨ r)
(p → q) ∧ (q → p)
1.33
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
The following proposition are logical equivalence? Prove it or give
an example?
a)
p ∧ (p → q)
b)
p→q
v
¬p ∨ (p ∧ q)
c)
p→q
v
¬p ∨ ¬q
d)
¬p
¬(p ∨ q) ∨ (¬p ∧ q)
e)
f)
v
v
p∧q
Contents
Propositional Logic
Logical Equivalences
Exercise
[(p ↔ q) ∧ (q ↔ r) ∧ (r ↔ p)]
[(p → q) ∧ (q → r) ∧ (r → p)]
v
[(p ∧ q) ∨ (q ∧ r) ∨ (r ∧ p)]
[(p ∨ q) ∧ (q ∨ r) ∧ (r ∨ p)]
v
1.34
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Determine the truth value and find the contrapositions as well as
the contradictions of the following propositions.
a) If ABCD is a rectangle, AB and CD are perpendicular.
b) If 14 is an odd number, 15 is divisible by 4.
c)
Contents
Two equal triangles have the same area.
d) If the quadratic equation
x
y
ax + bx + c = 0
e)
If two numbers
f)
If 45 ended with 5, 45 is divisible by 5.
g) If
√
2
and
2
are both divisible by
is an irrational number then
√ √
2. 2
has
a.c < 0,
n, (x + y)
it has root.
is also divisible by n.
Propositional Logic
Logical Equivalences
Exercise
is an irrational number.
h) If Pythagoras is French, Vietnam belongs to Asia.
3n + 2
is an odd integer,
n
i)
If
j)
If 8 < 9, 5 is a prime number.
is an odd integer.
k) A quadrilateral is a rhombus when it has 2 perpendicular diagonals.
l)
If 5 < 3, 7 is a prime number.
1.36
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Let
p
and
• p:
• q:
q
be:
Toan, Tran Hong Tai
"Brandon likes reading"
"Brandon is a good student"
The statement that formalize "If Brandon likes reading, Brandon
is a good student, vice versa, If Brandon is a good student,
Brandon like reading" is:
Contents
Propositional Logic
Logical Equivalences
Exercise
A)
(p ∧ q) → r
B)
p→q
D)
p∨q
p∧q
E)
p↔q
F)
¬p → ¬q
G)
¬p ∨ (p ∧ q)
H)
None of the others.
C)
1.38
Exercise
Logics
Nguyen An Khuong,
Let
P , Q, R
be:
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
• P:
• Q:
Potter is studying Math.
• R:
Potter is studying English.
Potter is studying Computer science.
Formalize the following statement using the propositional
connectives.
Example
Potter is studying Math and English but not Computer science:
Contents
Propositional Logic
Logical Equivalences
Exercise
P ∧ R ∧ ¬Q
a)
Potter is studying Math and Computer science but not
Computer science and English at the same time.
b)
It is not true that Potter is studying English and not Math.
c)
It is not true that Potter is studying English or Computer
science and not Math.
d)
Potter is not studying both Computer science and English but
is studying Math.
1.39
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Determine the wrong statement among the following.
a)
x ∈ {x}
b)
{x} ⊆ {x}
c)
{x} ∈ {x}
Contents
d)
{x} ∈ {{x}}
Propositional Logic
e)
∅ ⊆ {x}
A)
a
B)
b
C)
c
D)
d
E)
none of the others.
Logical Equivalences
Exercise
1.40
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
Which of the following proposition is a truth.
Contents
A)
(p ∨ ¬q) → q
Propositional Logic
B)
p → (p ∧ q)
Logical Equivalences
C)
¬p → (p → q)
D)
¬(p → q) → q
E)
none of the others.
Exercise
1.41
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Let's consider a propositional language where:
• p: ABC
• q : ABC
• r: ABC
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
is an isosceles triangle .
is an equilateral triangle .
has a
60o
angle .
Contents
Propositional Logic
Which of the following compounds formalize the theorem: if
ABC
is an isosceles triangle and has a
equilateral triangle
60o
angle then it is an
Logical Equivalences
Exercise
?
A)
(p ∧ q) → r
B)
(p ∧ r) → q
C)
(p ∧ r) ∨ q
D)
q → (p ∨ r)
E)
none of the others.
1.42
Exercise
Logics
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
There are 6 soccer teams A, B, C, D, E, F contested in a
tournament. The following are statements on which two teams are
in the grand final:
a.
A and C
b.
B and E
c.
B and F
d.
A and F
e.
A and D
Contents
Propositional Logic
Logical Equivalences
Exercise
Knowing that there are 4 half true statements and 1 totally false
statement. What teams are in the grand final?
1.43
Exercise
Logics
Find the truth values of the following statements (with brief
Tien Thinh, Mai Xuan
Toan, Tran Hong Tai
a) ∀x
∈ N, x2 + 5x + 6
b) ∃x
∈ R, x2 + x + 1 ≤ 0
c)
∃n
d) ∀n
∈ N, (n3 − n)
∈ N ∗, n2 − 1
e)
∀x, ∀y
f)
∃r
Nguyen An Khuong,
Tran Tuan Anh, Nguyen
explanations):
is not a prime number.
is not a multiple of 3.
is a multiple of 3.
∈ R, x2 + y 2 > 2xy ∈ Q, 3 < r < π Contents
g) ∃n
∈ N, n2 + 1
h) ∀x
∈ R, |x| < 3 ⇔ x2 < 9
divisible by 8
Propositional Logic
Logical Equivalences
∈ R, (a + b)2 > 2(a2 + b2 )
i)
∃a, b
j)
All real numbers are positive.
Exercise
k) There is a liquid metal.
l)
All equilateral triangles are equal.
m) All gases are non-conductive.
n) There exist quadrilaterals which don't have circumcircles.
o) There is a natural number n that, for all real numbers
f (x) = x2 − 2x + n is not negative.
p) For all positive integers
x
q) For all positive integers
x,
and
y
we have
x
There is a positive integer
s)
There exist positive integers
y
so that
that, for all positive integers
x
and
y
we have
x ≤ y .
there is a positive integer
r)
x,
so that
y,
x ≤ y .
we have
x ≤ y .
x ≤ y .
1.45