Chapter 1 Logics Discrete Mathematics I on 13 March 2012

Logics
Tran Vinh Tan

Chapter 1
Logics
Discrete Mathematics I on 13 March 2012

Contents
Propositional Logic

Tran Vinh Tan
Faculty of Computer Science and Engineering
University of Technology - VNUHCM
1.1


Logics

Contents

Tran Vinh Tan

Contents
Propositional Logic

1 Propositional Logic

1.2


Logics

Logic

Tran Vinh Tan

Definition (Averroes)

The tool for distinguishing between the true and the false.
Contents
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


Logics

Applications in Computer Science

Tran Vinh Tan

• Design of computer circuits

Contents
Propositional Logic

• Construction of computer programs
• Verification of the correctness of programs
• Constructing proofs automatically
• Artificial intelligence
• Many more...

1.4


Logics

Propositional Logic

Tran Vinh Tan

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 Viet Nam.

• New York City is the capital of USA.
• 1+1=2
• 2+2=3

1.5


Logics

Examples

Tran Vinh Tan

Examples (Which of these are propositions?)
• How easy is logic!

Contents
Propositional 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

Logics

Notations

Tran Vinh Tan

Contents
Propositional Logic

• Propositions are denoted by p, q, . . .
• The truth value (”chân trị”) is true (T) or false (F)

1.7


Logics

Operators

Tran Vinh Tan

Negation - ”Phủ định”: ¬p

Contents
Propositional Logic

Bảng: Truth Table for Negation

p

¬p

T
F

F
T

1.8


Logics

Operators

Tran Vinh Tan

Conjunction - ”Hội”: p ∧ q
“p and q”

Disjunction - ”Tuyển”: p ∨ q
“p or q”

Contents
Propositional Logic

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.

We need students who have
experience in Java or C++.
Tomorrow, I will eat Pho or Bun
bo.

1.9


Logics

Operators

Tran Vinh Tan

Exclusive OR - Tuyển loại: p ⊕ q
“p or q (but not both)”

Implication - Kéo theo: p → q
“if p, then q”

p

q

p⊕q

p

q

p→q

T
T
F
F

T
F
T
F

F
T
T
F

T
T

F
F

T
F
T
F

T
F
T
T

Contents
Propositional Logic

If it rains, the pavement will be
wet.

1.10


Logics

More Expressions for Implication p → q

Tran Vinh Tan

• if p, then q
• p implies q

Contents
Propositional Logic

• 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


Logics

Conditional Statements From p → q

Tran Vinh Tan

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
Tran Vinh Tan

Exercise

What are the converse and contrapositive of the following
conditional statement
“If he plays online games too much, his girlfriend leaves him.”

Contents
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

Tran Vinh Tan

p↔q
“p if and only if q”
Contents

p

q

p→q

T
T
F
F

T
F
T
F

T
F
F
T

Propositional Logic

• “p is necessary and sufficient for q”.
• “if p then q, and conversely”.
• “p iff q”.

1.14


Logics

Translating Natural Sentences

Tran Vinh Tan

Exercise

I will buy a new phone only if I have enough money to buy iPhone
4 or my phone is not working.

Contents
Propositional Logic

• 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


Logics

Translating Natural Sentences

Tran Vinh Tan

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

Tran Vinh Tan

Exercise

Construct the truth table of the compound proposition
(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

Contents
Propositional Logic

1.17


Logics

Applications

Tran Vinh Tan

Contents

• System specifications
• “When a user clicked on Help button, a pop-up will be shown
up”

Propositional Logic

• Boolean search
• type “dai hoc bach khoa” in Google
• means “dai AND hoc AND bach AND khoa”

1.18


Logics

Applications (cont.)

Tran Vinh Tan

• 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”?

Contents
Propositional Logic

• Bit operations
• 101010011 is a bit string of length nine.

1.19


Logics

Tautology and Contradiction

Tran Vinh Tan

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


Logics

Logical Equivalences

Tran Vinh Tan

Contents

Definition

Propositional Logic

The compound compositions p and q are called logically equivalent
if p ↔ q is a tautology, denoted p ≡ q.
Example

Show that ¬(p ∨ q) and ¬p ∧ ¬q are logically equivalent.

1.21


Logics

Logical Equivalences

Tran Vinh Tan

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

Contents
Propositional Logic

1.22


Logics

Logical Equivalences

Tran Vinh Tan

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

Tran Vinh Tan

Equivalence
p ∨ ¬p
p ∧ ¬p
(p → q) ∧ (p → r)
(p → r) ∧ (q → r)

(p → q) ∨ (p → r)
(p → r) ∨ (q → r)
p↔q

Contents

≡
≡
≡
≡
≡
≡
≡

Propositional Logic

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

Tran Vinh Tan

Example

Show that ¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent by
developing a series of logical equivalences.
Contents

Solution
¬(p ∨ (¬p ∧ q))

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 ∧ p ≡ F

≡

¬p ∧ ¬q

by the identity law for F

Consequently, ¬(p ∨ (¬p ∧ q)) and ¬p ∧ ¬q are logically equivalent.

1.25