Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Chapter 1
Logics
Contents
Propositional Logic
Discrete Structures for Computer Science (CO1007) on Ngày
13 tháng 1 năm 2017
Nguyen An Khuong, Huynh Tuong Nguyen
Faculty of Computer Science and Engineering
University of Technology, VNU-HCM
1.1
Contents
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
1 Propositional Logic
1.2
Logic
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Definition (Averroes)
Contents
The tool for distinguishing between the true and the false.
Propositional Logic
Definition (Penguin Encyclopedia)
The formal systematic study of the principles of valid inference
and correct reasoning.
Definition (Discrete Mathematics - Rosen)
Rules of logic are used to distinguish between valid and invalid
mathematical arguments.
1.3
Applications in Computer Science
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
• Design of computer circuits
• Construction of computer programs
• Verification of the correctness of programs
• Constructing proofs automatically
• Artificial intelligence
• Many more...
1.4
Propositional Logic
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Definition
Contents
Propositional Logic
A proposition is a declarative sentence that is either true or false,
but not both.
Examples
• Hanoi is the capital of Viet Nam.
• New York City is the capital of USA.
• 1+1=2
• 2+2=3
1.5
Examples
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Examples (Which of these are propositions?)
Contents
Propositional Logic
• How easy is logic!
• Read this carefully.
• H1 building is in Ho Chi Minh City.
• 4>2
• 2n ≥ 100
• The sun circles the earth.
• Today is Thursday.
• Proposition only when the time is specified
1.6
Notations
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
• Propositions are denoted by p, q, . . .
• The truth value (”chân trị”) is true (T) or false (F)
1.7
Logics
Operators
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Negation - ”Phủ định”: ¬p
Propositional Logic
Bảng: Truth Table for Negation
p
¬p
T
F
F
T
1.8
Logics
Operators
Nguyen An Khuong,
Huynh Tuong Nguyen
Conjunction - ”Hội”: p ∧ q
“p and q”
Disjunction - ”Tuyển”: p ∨ q
“p or q”
p
q
p∧q
p
q
p∨q
T
T
F
F
T
F
T
F
T
F
F
F
T
T
F
F
T
F
T
F
T
T
T
F
I’m teaching DM1 and it is
raining today.
Contents
Propositional Logic
We need students who have
experience in Java or C++.
Tomorrow, I will eat Pho or Bun
bo.
1.9
Logics
Operators
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Exclusive OR - Tuyển loại tru:
p⊕q
“p or q (but not both)”
p
q
p⊕q
T
T
F
F
T
F
T
F
F
T
T
F
Implication - Kéo theo: p → q
“if p, then q”
p
q
p→q
T
T
F
F
T
F
T
F
T
F
T
T
Propositional Logic
If it rains, the pavement will be
wet.
1.10
More Expressions for Implication p → q
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
• if p, then q
Propositional Logic
• p implies q
• p is sufficient for q
• q if p
• p only if q
• q unless ¬p
• If you get 100% on the final, you will get 10 grade.
• If you feel asleep this afternoon, then 2 + 3 = 5.
1.11
Conditional Statements From p → q
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
• q → p (converse - đảo)
• ¬q → ¬p (contrapositive - phản đảo)
• Prove that only contrapositive have the same truth table with
p→q
1.12
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Exercise
Contents
What are the converse and contrapositive of the following
conditional statement
“If he plays online games too much, his girlfriend leaves him.”
Propositional Logic
• Converse: If his girlfriend leaves him, then he plays online
games too much.
• Contrapositive: If his girlfriend does not leave him, then he
does not play online games too much.
1.13
Logics
Biconditionals
Nguyen An Khuong,
Huynh Tuong Nguyen
p↔q
“p if and only if q”
Contents
Propositional Logic
p
q
p↔q
T
T
F
F
T
F
T
F
T
F
F
T
• “p is necessary and sufficient for q”.
• “if p then q, and conversely”.
• “p iff q”.
1.14
Translating Natural Sentences
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Exercise
Propositional Logic
I will buy a new phone only if I have enough money to buy iPhone
4 or my phone is not working.
• p: I will buy a new phone
• q: I have enough money to buy iPhone 4
• r: My phone is working
• p → (q ∨ ¬r)
1.15
Translating Natural Sentences
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
Exercise
He will not run the red light if he sees the police unless he is too
risky.
1.16
Logics
Construct Truth Table
Nguyen An Khuong,
Huynh Tuong Nguyen
Exercise
Contents
Construct the truth table of the compound proposition
(p ∨ ¬q) → (p ∧ q).
Propositional Logic
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
1.17
Applications
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
• System specifications
• “When a user clicked on Help button, a pop-up will be shown
up”
• Boolean search
• type “dai hoc bach khoa” in Google
• means “dai AND hoc AND bach AND khoa”
1.18
Applications (cont.)
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
• Logic puzzles
• There are two kinds of inhabitants on an island, knights, who
always tell the truth, and their opposites, knaves, who always
lie. You encounter two people A and B. What are A and B if
A says “B is a knight” and B says ”The two of us are
opposite types”?
• Bit operations
• 101010011 is a bit string of length nine.
1.19
Tautology and Contradiction
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Definition
A compound proposition that is always true (false) is called a
tautology (contradiction).
Contents
Propositional Logic
• Tautology: hằng đúng
• Contradiction: mâu thuẫn
Example
• p ∨ ¬p (tautology)
• p ∧ ¬p (contradiction)
1.20
Logical Equivalences
Logics
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
Definition
The compound compositions φ and ψ are called logically
equivalent if φ ↔ ψ is a tautology, denoted φ ≡ ψ.
Example
Show that ¬(p ∨ q) and ¬p ∧ ¬q are logically equivalent.
1.21
Logics
Logical Equivalences
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
p∧T
p∨F
≡
≡
p
p
Identity laws
Luật đồng nhất
p∨T
p∧F
≡
≡
T
F
Domination laws
Luật nuốt
p∨p
p∧p
≡
≡
p
p
Idempotent laws
Luật lũy đẳng
¬(¬p)
≡
p
Double negation law
Luât phủ định kép
Propositional Logic
1.22
Logics
Logical Equivalences
Nguyen An Khuong,
Huynh Tuong Nguyen
p∨q
p∧q
≡
≡
q∨p
q∧p
(p ∨ q) ∨ r
(p ∧ q) ∧ r
≡
≡
p ∨ (q ∨ r)
p ∧ (q ∧ r)
Associative laws
Luật kết hợp
p ∨ (q ∧ r)
p ∧ (q ∨ r)
≡
≡
(p ∨ q) ∧ (p ∨ r)
(p ∧ q) ∨ (p ∧ r)
Distributive laws
Luật phân phối
¬(p ∧ q)
¬(p ∨ q)
≡
≡
¬p ∨ ¬q
¬p ∧ ¬q
De Morgan’s law
Luật De Morgan
p ∨ (p ∧ q)
p ∧ (p ∨ q)
≡
≡
p
p
Commutative laws
Luật giao hoán
Contents
Propositional Logic
Absorption laws
Luật hút thu
1.23
Logics
Logical Equivalences
Nguyen An Khuong,
Huynh Tuong Nguyen
Contents
Propositional Logic
Equivalence
p ∨ ¬p
p ∧ ¬p
(p → q) ∧ (p → r)
(p → r) ∧ (q → r)
(p → q) ∨ (p → r)
(p → r) ∨ (q → r)
p↔q
≡
≡
≡
≡
≡
≡
≡
T
F
p → (q ∧ r)
(p ∨ q) → r
p → (q ∨ r)
(p ∧ q) → r
(p → q) ∧ (q → p)
1.24
Logics
Constructing New Logical Equivalences
Nguyen An Khuong,
Huynh Tuong Nguyen
Example
Show that ¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent by
developing a series of logical equivalences.
Contents
Propositional Logic
Solution
¬(p ∨ (¬p ∧ q))
≡
¬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 ∧ p ≡ F
≡
¬p ∧ ¬q
by the identity law for F
Consequently, ¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent.
1.25