Mathematics in the
Modern World
Chapter 2
Elementary
Mathematical Logic
Logic is the basis of all mathematical statements
and arguments.
Its basic rules are essential to understand
mathematical statements
“There exists an integer 𝑛 such that 𝑛2 = 𝑛"
“If 𝑥 is an integer, then 𝑥 + 1 is also an integer
“For all real number 𝑥 with 𝑥 < 0, 𝑥 2 > 0
3
Proposition
4
■
A proposition is a declarative sentence that is
either true or false, but NOT both.
■
The truth or falsity of a statement is called its
truth value.
■
Letters are commonly used to denote
propositional variables such as 𝒑, 𝒒, 𝒓, 𝒂𝒏𝒅 𝒔.
Example: Which of the following is a proposition?
Proposition
1.
The sun is the center of the solar system.
2.
Kindly give me some sushi. Not a Proposition
3.
What is the title of your favorite song?
4.
Kobe Bryant is a tall person.
Not a Proposition
Not a Proposition
5
4. Kobe Bryant is a tall person.
HE IS TALL.
HE IS NOT
TALL.
The statement is true and false at the
same time!
6
A paradox is a declarative sentence which could neither be
true, nor false, or is both true or false.
Types of Proposition
7
❑
A proposition is called simple if it contains
only one idea.
❑
A proposition is called compound if it is
composed of at least two simple
propositions joined by logical connectives.
Examples: Simple & Compound Propositions
1. The sun is the center of the solar system.
Simple
2. Heart Evangelista is the wife of Sen. Chiz Escudero.Simple
3. Pres. Duterte is the 16th president while Vice Pres. Robredo
is either the 12th or the 14th vice president of the Republic of
the Philippines.
Compound
4. If the books will arrive on Thursday, they will be distributed
to the students on Friday.
Compound
8
Basic Logical
Operators
Operator
Meaning
Symbolic Form
Read as:
Negation
Not
~𝑝
“Not 𝑝”
Conjunction
And
𝑝∧𝑞
“𝑝 and 𝑞”
Disjunction
Or
𝑝∨𝑞
“𝑝 or 𝑞”
Conditional
If… then…
𝑝→𝑞
“If 𝑝, then 𝑞.”
Biconditional
If and only if
𝑝⟷𝑞
“𝑝 if and only if
𝑞.”
Negations: ~𝑝
This logical operator states the exact opposite of a given
statement
■
𝑝: The sum of two odd numbers is even
■
~𝑝: The sum of two odd numbers is not even
■
𝑞: It is sunny today
■
~𝑞: It is not sunny today.
𝒑
T
F
∼𝒑
F
T
Conjunction: 𝑝 ∧ 𝑞
We observe the connectives and, but, moreover, while
■
𝑝: 25 is a perfect square number
■
𝑞: 5 is one of the square roots of 25
■
𝑝 ∧ 𝑞: 25 is a perfect square number while 5 is one
of its square roots
𝒑
T
𝒒
T
𝒑∧𝒒
T
T
F
F
■
𝑝 : It is sunny today
■
𝑞: I need to go to work.
F
T
F
■
𝑝 ∧ 𝑞: It is sunny today and I need to go to work
F
F
F
Disjunction: 𝑝 ∨ 𝑞
We observe the words or, unless, either-or, etc.
■
𝑝: Mary will buy her clothes at the mall
■
𝑞: Mary will go to church on Sunday
■
𝑝 ∨ 𝑞: Either Mary will buy her clothes at the mall or
she will go to church on Sunday
𝒑
T
𝒒
T
𝒑∨𝒒
T
T
F
T
■
𝑝 : It is sunny today
F
T
T
■
𝑞: I need to go to work.
■
𝑝 ∨ 𝑞: It is sunny today or I need to go to work
F
F
F
Conditional: 𝑝 → 𝑞
We observe the words if-then, only if, implies, provided
that, given that, etc.
■ In 𝑝 → 𝑞, we say that “𝑝 is a sufficient condition for 𝑞”
while “𝑞 is a necessary condition for 𝑝”.
■ 𝑝 is called the antecedent of the implication.
■ 𝑞 is called the consequent of the implication
■
𝑝: The curfew is effective today
■ 𝑞: The police officers will arrest lawbreakers
■
■
𝑝 → 𝑞: If the curfew is effective today, then the police
officers will arrest lawbreakers
Conditional: 𝑝 → 𝑞
■
𝑝: 𝑛 is a negative integer.
■
𝑞: 𝑛2 is a positive integer.
■
𝑝 → 𝑞: If 𝑛 is a negative integer then 𝑛2 is a
positive integer.
■
In an implication 𝑝 → 𝑞, it means that if 𝑝 is
true, then 𝑞 is also true.
■
However, an implication is directional, i.e., if 𝑄
is true, it does not necessarily follow that 𝑃 is
also true.
𝒑
T
𝒒
T
𝒑→𝒒
T
T
F
F
F
T
T
F
F
T
Biconditional: 𝑝 ↔ 𝑞
We observe the words if and only if, is a necessary and
sufficient condition, etc.
■
𝑝: I will graduate on time.
■
𝑞: I will pass all the subjects this semester.
■
𝑝 ↔ 𝑞: I will graduate on time if and only if I will pass all the
subjects this semester.
■
𝑝: The tournament will take place tomorrow.
■
𝑞: The weather is good.
■
𝑝 ↔ 𝑞: The tournament will take place tomorrow if and only
if the weather is good.
Biconditional: 𝑝 ↔ 𝑞
If 𝑝 ↔ 𝑞 is true, we say that “𝑝 and
𝑞 are logically equivalent”.
■ That is, they will be true under
exactly the same circumstances.
■
𝒑
T
𝒒
T
𝒑↔𝒒
T
T
F
F
F
T
F
F
F
T
Translate each of the following statements into its
symbolic form
Upon announcement of Public Storm Warning Signal No. 3, classes
in all levels should be suspended and children should stay inside
strong buildings.
𝑃: Public Storm Warning Signal no. 3 is announced.
𝑄: Classes in all levels are suspended.
𝑅: Children should stay inside strong buildings.
𝑷 → (𝑸 ∧ 𝑹)
18
Translate each of the following statements into its
symbolic form
Two lines A and B are parallel if and
only if they are coplanar and they do
not intersect.
𝑃: Two lines A and B are parallel.
𝑄: Lines A and B are coplanar.
𝑅: Lines A and B intersect.
𝑷 ↔ (𝑸 ∧ ~𝑹)
19
TRUTH TABLES of
COMPOUND PROPOSITIONS
A TRUTH TABLE is a table showing all possible truth
values of a particular compound proposition.
21
CONSTRUCT THE TRUTH TABLES OF
THE GIVEN COMPOUND
PROPOSITION:
𝒑 ∧ ~𝒑
𝒑
~𝒑
𝒑 ∧ ~𝒑
T
F
F
F
T
F
Contradiction
22
CONSTRUCT THE TRUTH TABLES OF
THE GIVEN COMPOUND
PROPOSITION:
23
(𝒑 → 𝒒) ∨ (𝒒 → 𝒑)
(𝒒 → 𝒑) (𝒑 → 𝒒) ∨ (𝒒 → 𝒑)
T
T
𝒑
𝒒
(𝒑 → 𝒒)
T
T
T
T
F
F
T
F
F
T
F
T
F
T
T
T
T
T
Tautology
CONSTRUCT THE TRUTH TABLES OF THE GIVEN
COMPOUND PROPOSITION:
24
𝒑
𝒒
𝒓
~𝒒
(𝒑 ∧ ~𝒒)
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
F
F
T
T
F
F
T
T
F
F
T
T
F
F
F
F
(𝒑 ∧ ~𝒒) → 𝒓
𝒑 ∧ ~𝒒 → 𝒓
T
T
T
F
T
T
T
T
Contingency
Tautology, Contradiction, and Contingency
A contradiction is a compound proposition that is false for all
possible truth values of its component propositions.
A tautology is a compound proposition that is true for all
possible truth values of its component propositions.
A contingency is a compound proposition that is neither a
tautology nor a contradiction.
25
Logical
Equivalence
26
Logical Equivalence ≡
Given any compound propositions 𝑝 and 𝑞, 𝑝 is said to be
logically equivalent to 𝑞
if 𝒑 ↔ 𝒒 is a tautology. It is denoted by 𝑝 ≡ 𝑞 .
𝒑
T
T
F
F
27
𝒒
T
F
T
F
~𝒑
F
F
T
T
~𝒒
F
T
F
T
𝒑→𝒒
T
F
T
T
~𝒑 ∨ 𝒒
T
F
T
T
~𝒒 → ~𝒑
T
F
T
T
Predicates and
Quantifiers
28
A predicate is a sentence containing variables. Its truth value
depends on the values to be assigned on the variables
indicated.
𝑥 is a real number
𝑦 + 2 = 10
𝑥−𝑦>𝑧
A predicate is also known as a propositional function usually denoted
by P(x), Q(x,y), R(x,y,z), etc.
A counter example is a value assigned to a variable for which
the statement is false.
29
■
Once a value is assigned to variables, the predicate
becomes a proposition with its corresponding truth
value.
■
However, there is another method to transform
predicates to propositions. This is where quantifiers
are introduced.
■
30
Quantifiers refer to the amount to which a predicate is
true over a particular domain of discourse or simply, the
domain, which is the set of values for which the
statement is defined.
Types of Quantifiers
Universal quantifier, denoted by ∀,
refers to the phrase “for all” or “for
every” or “for each”
∀𝑥 ∈ 𝑅, 𝑥 2 ≥ 0
❑ ∀𝑥 ∈ 𝑁, 𝑥 − 5 > 0
❑ ∀𝑥, ∀𝑦 ∈ 𝑅, 𝑥 + 𝑦 = 𝑦 + 𝑥
❑
Universal quantifier
True 1) ∀𝑥 ∈ 𝑅, 𝑥 2 ≥ 0
False 2) ∀𝑥 ∈ 𝑁, 𝑥 − 5 > 0
True 3) ∀𝑥, ∀𝑦 ∈ 𝑅, 𝑥 + 𝑦 = 𝑦 + 𝑥
32
Counterexample:
𝑤ℎ𝑒𝑛 𝑥 = 3
𝑥−5>0
3−5>0
−2 > 0
𝐹𝑎𝑙𝑠𝑒
Types of Quantifiers
Existential quantifier, denoted by
∃, refers to the phrase “there
exists” or “for at least one” or “for
some”
∃𝑥 ∈ ℝ, 𝑥 2 − 10 ≥ 0
❑ ∃𝑥 ∈ ℕ, 𝑥 − 5 < 0
❑ ∃𝑥 ∈ ℤ, 3𝑥 + 2 = 6
❑
Existential quantifier
True 1) ∃𝑥 ∈ ℝ, 𝑥 2 − 10 ≥ 0
True 2) ∃𝑥 ∈ ℕ, 𝑥 − 5 < 0
False 3) ∃𝑥 ∈ ℤ, 3𝑥 + 2 = 6
34
3𝑥 + 2 = 6
3𝑥 = 6 − 2
3𝑥 = 4
4
𝑥=
3
4
∉ℤ
3
END
of
Chapter 2
35