Term 1, S.Y.

2023-2024 De La Salle University

Taft Avenue, Manila

GEMATMW:
Mathematics in the
Modern World
Von Anthony G. Torio, Ph.D.
von.torio@dlsu.edu.ph

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

The Course Contents

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Course Topics
Module # Topics

1 Mathematics as a Language

2 Mathematics in Decision Sciences

3 Mathematics in Natural Sciences and Arts

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Course Topics
Module # Topics

4 Mathematics for Ef ciency

5 Mathematics for Digital Communications

6 Mathematics in Finance

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
fi
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Mathematics as a
Language
Essential Questions
1. How can I view Mathematics as a
fundamental tool applicable to a wide range
of disciplines?
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y.
2022-2023 De La Salle University
Taft Avenue, Manila

Course Tasks
Collaborative Task
Adjusted Due: September 20,
2023
@11:59 pm
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y.
2022-2023 De La Salle University
Taft Avenue, Manila

Course Tasks
Individual Work

Due: September 13, 2023

@11:59 pm
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y.
2022-2023 De La Salle University
Taft Avenue, Manila

Course Tasks
Individual Work

Due: September 20, 2023

@11:59 pm
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Mathematics as a
Language
Topics:

1. Logic and Reasoning

2. Mathematical Reasoning
Adapted from the video
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
presentation of Nocon, 2021
Von Anthony G. Torio, Ph.D.
Term 1, S.Y.
2023-2024 De La Salle University
Taft Avenue, Manila

GEMATMW: Mathematics in the Modern World/ DLSU - Manila Photo by Christopher Sardegna on unsplash.com
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Mathematical
Reasoning
★Refers to the ability of a
person to analyse problem
situations and construct logical
arguments to create both
conceptual foundations and
connections to be able to
process the available
information and solve the
problems.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Inductive vs Deductive
★Inductive reasoning -
process of making general
conclusions based on specific
examples. Example:
Every object I release falls to
the ground. Therefore, the next
object I will release will fall to
the ground.
ISG - Inductive (Specific to General)
DGS - Deductive (General to Specific)
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Inductive vs Deductive
★Deductive reasoning -
process of making specific
conclusions based on general
principles. Example:
All men are mortal. I am a
man. Therefore, I am mortal.
General Principle: If p implies q
and q holds, then q must follow.
ISG - Inductive (Specific to General)
DGS - Deductive (General to Specific)
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Definitions
★A theorem is a statement that can be shown to be true.
★It is formulated by using a sequence of statements that form an
argument, called proof.
★The statements used in a proof may include axioms - underlying
assumptions about mathematical structures, the hypothesis of the
theorem (scientific guess), and previously proven theorems.
★ The rules of inference tie together the steps of a proof.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

Addition
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

Simplification
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

Conjunction
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

Modus ponens
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

Modus tollens
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

Hypothetical syllogism
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

Deductive syllogism
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Examples

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

I will either drink cola or melon juice. I am

not drinking cola. Therefore, I am drinking
melon juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

I will either drink

cola or melon juice.
I am not drinking
cola. Therefore, I
am drinking melon
juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

I will either drink

cola or melon juice.
I am not drinking
cola. Therefore, I
am drinking melon
juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

I will either drink

cola or melon juice.
I am not drinking
cola. Therefore, I
am drinking melon
juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

I will either drink cola or melon juice. I am

not drinking cola. Therefore, I am drinking
melon juice.

Answer: Disjunctive Syllogism

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2

Anna is a human resource management

major. Therefore, Anna is either a human
resource management major or a computer
applications major.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2
Anna is a human
resource management
major. Therefore,
Anna is either a
human resource
management major or
a computer
applications major.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2
Anna is a human
resource management
major. Therefore,
Anna is either a
human resource
management major or
a computer
applications major.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2
Anna is a human
resource management
major. Therefore,
Anna is either a
human resource
management major or
a computer
applications major.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2

Anna is a human resource management

major. Therefore, Anna is either a human
resource management major or a computer
applications major.

Answer: Addition

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3

Ben is a game designer and a game

developer. Therefore, Ben is a game
designer.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3

Ben is a game
designer and a
game developer.
Therefore, Ben is a
game designer.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3

Ben is a game
designer and a
game developer.
Therefore, Ben is a
game designer.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3

Ben is a game
designer and a
game developer.
Therefore, Ben is a
game designer.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3

Ben is a game designer and a game

developer. Therefore, Ben is a game
designer.

Answer: Simplification

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 4

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 4

If it rains today, the college will be closed.

The college is not closed today. Therefore,
it did not rain today.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 4

If it rains today, the

college will be
closed. The college
is not closed today.
Therefore, it did not
rain today.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 4

If it rains today, the

college will be
closed. The college
is not closed today.
Therefore, it did not
rain today.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 4

If it rains today, the

college will be
closed. The college
is not closed today.
Therefore, it did not
rain today.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 4

If it rains today, the college will be closed.

The college is not closed today. Therefore,
it did not rain today.

Answer: Modus Tollens

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 5

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 5

If it is rainy, the oval will be closed. It is

rainy. Therefore, the oval is closed.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 5

If it is rainy, the oval

will be closed. It is
rainy. Therefore, the
oval is closed.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 5

If it is rainy, the oval

will be closed. It is
rainy. Therefore, the
oval is closed.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 5

If it is rainy, the oval

will be closed. It is
rainy. Therefore, the
oval is closed.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 5

If it is rainy, the oval will be closed. It is

rainy. Therefore, the oval is closed.

Answer: Modus Ponens

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 6

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 6

If I go swimming, then I will stay in the sun too

long. If I stay in the sun too long, then I will
get burned. Therefore, if I go swimming, then I
will get burned.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 6

If I go swimming,
then I will stay in the
sun too long. If I
stay in the sun too
long, then I will get
burned. Therefore, if
I go swimming, then
I will get burned.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 6

If I go swimming,
then I will stay in the
sun too long. If I
stay in the sun too
long, then I will get
burned. Therefore, if
I go swimming, then
I will get burned.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 6

If I go swimming,
then I will stay in the
sun too long. If I
stay in the sun too
long, then I will get
burned. Therefore, if
I go swimming, then
I will get burned.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 6

If I go swimming, then I will stay in the sun too

long. If I stay in the sun too long, then I will
get burned. Therefore, if I go swimming, then I
will get burned.

Answer: Hypothetical Syllogism

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 7

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 7

I will eat pizza. I will drink orange juice.

Therefore, I will eat pizza and drink orange
juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 7

I will eat pizza. I will

drink orange juice.
Therefore, I will eat
pizza and drink
orange juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 7

I will eat pizza. I will

drink orange juice.
Therefore, I will eat
pizza and drink
orange juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 7

I will eat pizza. I will

drink orange juice.
Therefore, I will eat
pizza and drink
orange juice.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 7

I will eat pizza. I will drink orange juice.

Therefore, I will eat pizza and drink orange
juice.

Answer: Conjunction

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Fallacies

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Fallacies
★Arguments constructed using the rules of inference are said to be
valid.
★When all propositions used in a valid argument are true, it leads to
a correct conclusion.
★Fallacies are incorrect reasoning which appear to follow the rules of
inference but are based on contingencies rather than tautologies.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Tautology

A compound proposition that is always true, no matter

what the truth values of the propositions that occur in it
are, is called a tautology.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Contradiction

A compound proposition that is always false, no matter

what the truth values of the propositions that occur in it
are, is called a contradiction.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Contingency

A compound proposition that is neither a tautology nor

a contradiction is called a contingency.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Three Types of Fallacies (Summary)

1. The fallacy of affirming the conclusion
- Based on the compound proposition:

2. The fallacy of denying the hypothesis

- Based on the compound proposition:

3. Begging the question or circular reasoning

- Occurs when one or more steps of a proof are
based on the truth of the statement being proved.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Three Types of Fallacies

1. The fallacy of affirming the conclusion
- Based on the compound proposition:

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

If you do every problem in a math book, then

you will learn Mathematics. You learned
Mathematics. Therefore, you did every
problem in the book.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1
If you do every
problem in a math
book, then you will
learn Mathematics. You
learned Mathematics.
Therefore, you did
every problem in the
book.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1
If you do every
problem in a math
book, then you will
learn Mathematics. You
learned Mathematics.
Therefore, you did
every problem in the
book.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1
If you do every
problem in a math
book, then you will
learn Mathematics. You
learned Mathematics.
Therefore, you did
every problem in the
book.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Alternatively…

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1
If you do every
problem in a math
book, then you will
learn Mathematics. You
learned Mathematics.
Therefore, you did
every problem in the
book.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Three Types of Fallacies (Summary)

1. The fallacy of affirming the conclusion
- Based on the compound proposition:

2. The fallacy of denying the hypothesis

- Based on the compound proposition:

3. Begging the question or circular reasoning

- Occurs when one or more steps of a proof are
based on the truth of the statement being proved.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1

If you do every problem in a math book, then

you will learn Mathematics. You learned
Mathematics. Therefore, you did every
problem in the book.

Answer: Fallacy of affirming the conclusion!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Three Types of Fallacies

2. The fallacy of denying the hypothesis
- Based on the compound proposition:

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2

If you do every problem in a Math book, then

you will learn Mathematics. You did not do
every problem in the book. Therefore, you did
not learn Mathematics.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2
If you do every
problem in a Math
book, then you will
learn Mathematics.
You did not do every
problem in the book.
Therefore, you did not
learn Mathematics.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

`
If you do every
problem in a Math
book, then you will
learn Mathematics.
You did not do every
problem in the book.
Therefore, you did not
learn Mathematics.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2
If you do every
problem in a Math
book, then you will
learn Mathematics.
You did not do every
problem in the book.
Therefore, you did not
learn Mathematics.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Alternatively…

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2
If you do every
problem in a math
book, then you will
learn Mathematics. You
learned Mathematics.
Therefore, you did
every problem in the
book.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Three Types of Fallacies (Summary)

1. The fallacy of affirming the conclusion
- Based on the compound proposition:

2. The fallacy of denying the hypothesis

- Based on the compound proposition:

3. Begging the question or circular reasoning

- Occurs when one or more steps of a proof are
based on the truth of the statement being proved.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2

If you do every problem in a Math book, then

you will learn Mathematics. You did not do
every problem in the book. Therefore, you did
not learn Mathematics.

Answer: Fallacy of denying the hypothesis!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Three Types of Fallacies

3. Begging the question or circular reasoning
- Occurs when one or more steps of a proof are based on the
truth of the statement being proved.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3
Consider proving the statement: “If n is
even, then n is
2
even,” is the following argument valid?
Suppose that n =
2k for some integer k. Let n = 2l for
2
some integer l. This shows that n is even.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Analysis

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3
Consider proving the
statement: “If n is even,
2
This is a circular
then n is even,” is the
following argument valid? argument (same as
Suppose that even. n is
2 the proposition
Then n = 2k for some
2
integer k. Let n = 2l for being proved).
some integer l. This
shows that n is even.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Three Types of Fallacies (Summary)

1. The fallacy of affirming the conclusion
- Based on the compound proposition:

2. The fallacy of denying the hypothesis

- Based on the compound proposition:

3. Begging the question or circular reasoning

- Occurs when one or more steps of a proof are
based on the truth of the statement being proved.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3
Consider proving the statement: “If even, then n isn is
2
even,” is the following argument valid?
Suppose that even. Then n is
2 n =
2k for some integer k. Let
2
n = 2l for some integer l. This shows that n is even.

Answer: Circular reasoning!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof that the

Vacuous implication p ⟹ q is true
Proof based on the fact that p
is false.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof that the

Trivial implication p ⟹ q is true
Proof based on the fact that q
is true.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof that the

Direct implication p ⟹ q is true
Proof by showing that q must
be true if p is true.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof that the

Indirect implication p ⟹ q is true
Proof by showing that p must
be false when q is false.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Can be used to prove implications (if, then statements)

p q p q
T T T
T F F
F T T
F F T
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Vacuous Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof that the

Vacuous implication p ⟹ q is true
Proof based on the fact that p
is false.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Just show that p is false!

p q p q
T T T
T F F
F T T
F F T
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 1
Vacuous Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Show that P(0) is true where P(n) is “If n>1,

then n >n.”
2

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Show that P(0) is true where P(n) is “If n>1,

then n > n.”
2

Vacuous Proof:
Note:
P(0) is “If 0 > 1, then 02 > 0.”

p is false!
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Show that P(0) is true where P(n) is “If n>1,

then n > n.”
2

Vacuous Proof:
Note:
P(0) is “If 0 > 1, then 02 > 0.”

The whole implication is true!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Trivial Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof that the

Trivial implication p ⟹ q is true
Proof based on the fact that q
is true.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Just show that q is true!

p q p q
T T T
T F F
F T T
F F T
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 2
Trivial Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Let P(n) be “ If a and b are positive

integers with , then .”
Show that P(0) is true.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Let P(n) be “If a and be are positive

integers with , then .”
Show that P(0) is true.
Trivial Proof:
P(0) is “If , then .”

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Let P(n) be “If a and be are positive

integers with , then .”
Show that P(0) is true.
Trivial Proof:
P(0) is “If , then .”

Since Hence, P(0) is true!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Direct Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof Follow Modus Ponens!

A proof that the

Direct implication p ⟹ q is true
Proof by showing that q must
be true if p is true.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Just show that p and q are true!

p q p q
T T T
T F F
F T T
F F T
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 3
Direct Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give a direct proof of the statement, “If n is

odd, then n is odd.”
2

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give a direct proof of the statement, “If n is

odd, then n is odd.”
2

Direct Proof:
Suppose n is odd. Then, n = 2k + 1 for some integer k.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give a direct proof of the statement, “If n is

odd, then n is odd.”
2

Direct Proof:
Suppose n is odd. Then, n = 2k + 1 for some integer k.
It follows that n = (2k + 1) = 4k + 4k + 1 = 2(2k + 2k) + 1.
2 2 2 2

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give a direct proof of the statement, “If n is

odd, then n is odd.”
2

Direct Proof:
Suppose n is odd. Then, n = 2k + 1 for some integer k.
It follows that n = (2k + 1) = 4k + 4k + 1 = 2(2k + 2k) + 1.
2 2 2 2

Therefore, n2 is odd (since it is 1 more than twice an integer).

GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Indirect Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof Follow Modus Tollens!

A proof that the

Indirect implication p ⟹ q is true
Proof by showing that p must
be false when q is false.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Rules of
Inference

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Just show that p and q are true!

p q p q
T T T
T F F
F T T
F F T
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 4
Indirect Proof

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give an indirect proof of the theorem, “If

3n + 2 is odd, then n is odd.”
Indirect Proof:
Assume that n is even. Then n = 2k for some integer k.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give an indirect proof of the theorem, “If

3n + 2 is odd, then n is odd.”
Indirect Proof:
Assume that n is even. Then n = 2k for some integer k.
It follows that 3n + 2 = 3 (2k) + 2 = 6k + 2 = 2(3k + 1).

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give an indirect proof of the theorem, “If

3n + 2 is odd, then n is odd.”
Indirect Proof:
Assume that n is even. Then n = 2k for some integer k.
It follows that 3n + 2 = 3 (2k) + 2 = 6k + 2 = 2(3k + 1).
Hence, 3n + 2 is even (since it is a multiple of 2).

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Give an indirect proof of the theorem, “If

3n + 2 is odd, then n is odd.”
Indirect Proof:
Assume that n is even. Then n = 2k for some integer k.
It follows that 3n + 2 = 3 (2k) + 2 = 6k + 2 = 2(3k + 1).
Hence, 3n + 2 is even (since it is a multiple of 2).
Negating the conclusion means that the hypothesis is false and the original implication is true.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Other Proofs

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof that the proposition

Proof by p is true based on the truth
contradiction of the implication - p ⟹ q
where q is a contradiction.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof of an implication where

the hypothesis is a disjunction of
Proof by
propositions showing that each
cases proposition separately implies
the conclusion.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof
A proof of a proposition of the form:

Existence proofs are classified as either constructive

Existence or non-constructive. Constructive – establishes the
assertion by exhibiting a value c such that P(c) is
proof true. Non-constructive – establishes the assertion
without indicating how to find a value x such that
P(x) is true. It commonly involves a proof by
contradiction.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

A proof of the statement:

Non-
existence One way is to assume that there is a
member of the universe of discourse for
proof which P(x) is true, and try to arrive at a
contradiction.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Methods of Proof

Let P(n) be a proposition for each

positive integer n. If the following two
Proof by conditions are satisfied, then P(n) is true
mathematical for all positive integers n: 1) The
induction proposition P(1) is true; and 2) The
implication P(n) ⟹ P(n + 1) is shown to
be true for every positive integer n.
GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 5
Proof by contradiction

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).
It follows that:

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).
It follows that: This implies that a2 is even.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).
It follows that: This implies that a2 is even.
Furthermore, since a is even, a = 2c for
GEMATMW: Mathematics in the Modern World/ DLSU - Manila some integer c.
Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).
It follows that: This implies that a2 is even.
Furthermore, since a is even, a = 2c for
some integer c.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).
It follows that: This implies that a2 is even.
Furthermore, since a is even, a = 2c for
some integer c.
Thus, 2b2 = 4c2, so that b2 = 2c2.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).
It follows that: This implies that a2 is even.
Furthermore, since a is even, a = 2c for
some integer c.
Thus, 2b2 = 4c2, so that b2 = 2c2.

This means that b2 is even. Hence, b must be even as well.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Proof by Contradiction
From this assumption, there exist integers a and b with
where a and b have no common factors (a/b is in lowest terms).
It follows that: This implies that a2 is even.
Furthermore, since a is even, a = 2c for
some integer c.
Thus, 2b2 = 4c2, so that b2 = 2c2.

This means that b2 is even. Hence, b must be even as well.

These imply that 2 divides a and b, contradicting the assumption that a and be
have no common factor.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 6
Proof by Constructive Existence

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Show that a square exists that is the sum

of two other squares.
Proof by Constructive existence

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Show that a square exists that is the sum

of two other squares.
Proof by Constructive existence
Consider 52.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Show that a square exists that is the sum

of two other squares.
Proof by Constructive existence
Consider 52.
Since 52 = 32 + 42, the proof is complete!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Example 7
Proof by Mathematical Induction

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila
Principle of Mathematical Induction

Prove that the sum of the rst n odd

positive integers is n .
2

Proof by Mathematical Induction

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
fi
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila
Principle of Mathematical Induction

Prove that the sum of the rst n odd

positive integers is n .
2

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
fi
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila
Principle of Mathematical Induction

Prove that the sum of the rst n odd

positive integers is n .
2

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
fi
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila
Principle of Mathematical Induction

Prove that the sum of the rst n odd

positive integers is n .
2

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12. TRUE!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
fi
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila
Principle of Mathematical Induction

Prove that the sum of the rst n odd

positive integers is n .
2

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12. TRUE!
Suppose P(n) is true for a positive integer n. That is:
GEMATMW: Mathematics in the Modern World/ DLSU - Manila 1 + 3 + 5 + … (2n - 1) = n2
Von Anthony G. Torio, Ph.D.
fi
Term 1, S.Y. 2023-2024

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12. TRUE!
Suppose P(n) is true for a positive integer n. That is:
1 + 3 + 5 + … (2n - 1) = n2

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12. TRUE!
Suppose P(n) is true for a positive integer n. That is:
1 + 3 + 5 + … (2n - 1) = n2
Assuming P(n) is true, it must be shown that P(n + 1) is true.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12. TRUE!
Suppose P(n) is true for a positive integer n. That is:
1 + 3 + 5 + … (2n - 1) = n2
Assuming P(n) is true, it must be shown that P(n + 1) is true.
1 + 3 + 5 + … (2n - 1) + [2(n + 1)-1] = (n + 1)2
n2 + (2n + 1) = (n + 1)2
n2 + 2n + 1 = n2 + 2n + 1

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12. TRUE!
Suppose P(n) is true for a positive integer n. That is:
1 + 3 + 5 + … (2n - 1) = n2
Assuming P(n) is true, it must be shown that P(n + 1) is true.
n2 + 2n + 1 = n2 + 2n + 1
Since the two conditions of the mathematical induction are satisfied,
it can be concluded that P(n) is true for all positive integers n.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024

Proof by Mathematical Induction

Let P(n) be the proposition: “The sum of the first n odd positive integers is n2.”

P(1): The sum of the first 1 odd positive integers is 12. TRUE!
Suppose P(n) is true for a positive integer n. That is:
1 + 3 + 5 + … (2n - 1) = n2
Assuming P(n) is true, it must be shown that P(n + 1) is true.
n2 + 2n + 1 = n2 + 2n + 1
Since the two conditions of the mathematical induction are satisfied,
it can be concluded that P(n) is true for all positive integers n.
That is, the sum of the first n odd positive integers is n.
2

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Thank you!

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Async next meeting.

GEMATMW: Mathematics in the Modern World/ DLSU - Manila

Von Anthony G. Torio, Ph.D.
Term 1, S.Y. 2023-2024 De La Salle University
Taft Avenue, Manila

Religio * Mores * Cultura

GEMATMW: Mathematics in the Modern World/ DLSU - Manila
Von Anthony G. Torio, Ph.D.