Name: Grade & Section:

Lesson 3
Propositions
In general, Logic is the art and science of right thinking.
“It is not about the reality of what a man thinks, but it is about the system of how a man
thinks.” Logic is all about the operation of thinking.
The first known documentation on logic were of Aristotle, which became the bases of modern
logic.
Propositional Logic is a branch of mathematical logic that studies the truth and falsity of
the given proposition.

Let’s Explore and Discover

A simple proposition is a declarative sentence for affirmative or denial. It is a statement
with truth value; either true (T) or false (F), but not both.
Statements Type of Has Truth Value? Proposition?
sentence
1. Napsan National High School Declarative Yes - True Propositions
is 53km away from the City
proper of Puerto Princesa.
2. Are you a Senior High School Interrogative No – Question Not
Student? What is your Propositions
Strand?
3. Find your LRN from your Imperative No – Command Not
card. Propositions
4. Welcome back to School! Exclamatory No – Expression Not
Propositions
5. I am a Grade 11 student living Declarative Yes – can be true or Propositions
in Puerto Princesa City. false based on the
person asked.
6. Triangle ABC is an isosceles Declarative No – no figure for Not
triangle. ΔABC Proposition
7. 2 + x = 18 Mathematical / Yes – can be yes or Propositions
Declarative no based on x value

Let’s Practice
Activity 1: Determine if each statement is a proposition.
1. All parallelograms are quadrilaterals.
2. Rhombuses are squares.
3. Is an equilateral triangle an isosceles triangle?
4. Triangle ABC (ΔABC) is a right triangle.
5. Draw two parallel lines that are cut by a transversal.

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Qualitative Categories of Propositions
Propositions can be categorized as affirmative and negative.
Propositions
Affirmative Negative

1. A quadrilateral has four sides. 1. A right triangle has no obtuse angle

2. The Philippines is a member of ASEAN. 2. Tomato is not a fruit.
3. Whales are mammals. 3. Parallel lines never intersects.

Let’s Practice
Activity 2: Which of the statements are affirmative and negative?
1. Antarctica is a desert.
2. No insects lived in the North pole.
3. Quotient is not the answer for addition.
4. The cerebrum constitutes 70% of the brain.
5. A triangle has no diagonal.

Quantitative Categories of Propositions

Categorical propositions are further classified according to quantity or the different
possible extensions of their subject terms.

Types Description Examples

Universal The subject term is taken • All quadrilaterals are polygons

Proposition in full extension. • No parallel lines meet at a point.
• Every integer is a real number.
Particular The subject term is taken • Some algebraic expressions are
Proposition only in particular polynomials.
extension.
Singular The subject term denotes • A prime number has only two
Proposition a single person or a thing. factors
• Only the triangle has no diagonal.

When quality and quantity are combined, propositions may be classified based on its
mood as follows: (where in x is the subject and y is the predicate)

Universal Particular

A I
Affirmative
All x are y Some x are y
E O
Negative All x are not y Some x are not y
No x are y Not all x are y

The letters A, E, I, and O can be used to refer to propositions universal affirmative,

universal negative, particular affirmative, and particular negative, respectively.

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 Let’s Practice
Activity 3: Determine whether each statement is A, E, I, and O proposition.
1. Every forest is a den of snakes.
2. Some crocodiles are found in the city.
3. Not all lamb is tame.
4. Some men are never as free as a birds.
5. No one is excused with the law.

Let’s Do This!
Activity 4: Complete the table below using A, E, I and O propositions.

A proposition E proposition I proposition O proposition

Universal Universal Particular Particular
Affirmative negative Affirmative Negative

1. All squares are

similar.

2. No SHS
students are
ABM students
in this school.
3. Some animals
are not found in
zoos.

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 Lesson 4
Syllogisms
Syllogisms are the cornerstone of mathematical logic, deals with hypothetical or
compound propositions.
A compound proposition is a proposition formed by combining two or more simple
propositions using logical operations and connectors.

Consider relating proposition 𝑝 to another proposition 𝑞 to form a new proposition.

Let’s Explore and Discover

Consider relating proposition 𝑝 to another proposition 𝑞 to form a new proposition.

Resulting Proposition
Symbol Read as
Proposition Symbolization

Λ and conjunction 𝑝∧𝑞

V or disjunction 𝑝∨𝑞

→ implies; If…, then… implication 𝑝 →𝑞

is equivalent to…
↔ equivalence 𝑝 ↔𝑞
If and only if…

~ Not negation ~𝑝

Logical Operators and Symbols:

1. Conjunction
The conjunction of proposition p and q. is denoted by p ∧ q (read as “p and q”);
combination of the statement.

Example: Express the of p ∧ q:

p: The moon is bright. p ∧ q: The moon is bright and
q: Annie is sleeping. Annie is sleeping.

Note that if the sentences are contrasting, you can use the word but instead of and.

~p ∧ q: The moon is not bright but Annie is sleeping.

2. Disjunction
The disjunction of propositions p and q is denoted by p ∨ q (read as “p or q”);

Example: Express the p ∨ q:

p: The moon is bright. p ∨ q: The moon is bright or Annie
q: Annie is sleeping. is sleeping

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3. Conditional/Implication
The conditional of propositions p and q is denoted by p → q: If p then q (read as “p
implies q”); While p is called hypothesis and q is the conclusion.

Example: Express the p → q:

p: The moon is bright. p → q: If the moon is bright then
q: Annie is sleeping. Annie is sleeping

4. Biconditional/Equivalence
The biconditional of propositions p and q is denoted by p ↔ q (read as “p if and
only if q”); can also be written as “p iff q”.

Example: Express the p ↔ q:

p: The moon is bright. p ↔ q: The moon is bright if and
q: Annie is sleeping. only if Annie is sleeping.

5. Negation
The negation of proposition p is denoted by ~p (read as “not p”). It is always the
opposite of the statement.

Example: Perform the negation of p and q:

p: The moon is bright. ~p: The moon is not bright.
q: Annie is sleeping. ~q: Annie is not sleeping.

If the whole compound proposition is negated, we use the phrase “It is not true
that…” or “It is not the case that…” at the beginning of the proposition.

Example:
p: The moon is bright. q: Annie is sleeping.

Perform the negation of each compound proposition:

~(p ↔ q): It is not the case that the moon is bright if and only if Annie is
sleeping.
~(q → p): It is not true that if the moon is bright then Annie is sleeping

Other Examples:
Let p, q and r be the following propositions:

p: Mariel is studying hard.

q: Patrick is playing basketball.
r: Lourince has a practice of cheerdance.

Perform the following logical operations.

Logical Operators Proposition in English Sentence
and Symbols
1. p ↔ q Mariel is studying hard if and only if Patrick is playing
basketball.
2. p ∨ q ∨ r Mariel is studying hard or Patrick is playing basketball or
Lourince has a practice of cheerdance.
3. p → q If Mariel is studying hard then Patrick is playing
basketball.
4. q ∧ r Patrick is playing basketball and Lourince has a practice
of cheerdance.
5. ~r Mariel is not studying hard
6. p → (r ∨ q) If Mariel is studying hard then Lourince has a practice of
cheerdance or Patrick is playing basketball.

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 7. p ∧ q ∨ r Mariel is studying hard and Patrick is playing basketball
or Lourince has a practice of cheerdance
8. (q ∧ r) → ~p If Patrick is playing basketball and Lourince has practice for
cheerdance, then Mariel is not studying hard.
9. ~q ∨ r Patrick is not playing basketball or Lourince has a
practice of cheerdance.
10. ~(p ∧ q) → ~p It is not the case that if Mariel is studying hard and Patrick
is playing basketball, then Mariel is not studying hard.

Let’s Practice
Activity 5: Write each compound sentence in symbolic form.
1. If the sky is cloudy, then probably it will rain. p: Probably it will rain.
2. It is not the case that the sky is cloudy if and only if
probably it will rain. q: The sky is cloudy.
3. Probably it will rain and the sky is cloudy
4. The sky is cloudy or probably it will not rain.
5. It is not true that if the sky is cloudy and probably it will not rain, then sky is not
cloudy

Let’s Do This!
Write each compound sentence in symbolic form.
Activity 6: Write each symbolism in ordinary English sentence.
p: Sir Kesh is a Math teacher. q: Sir Kesh is a UP student.

1. p ∨q
2. ~q ∧ p
3. p→q
4. ~q ↔ ~p
5. (~p ∨ q) → q
6. ~(p ↔ q)

Answer Key
Activity 1: Activity 2: Activity 3:
1. proposition 1. affirmative 1. A proposition
2. proposition 2. negative 2. I proposition
3. not proposition 3. negative 3. E proposition
4. not proposition 4. affirmative 4. O proposition
5. not proposition 5. negative 5. E proposition

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Activity 4:
A proposition E proposition I proposition O proposition
Universal Universal Particular Particular
Affirmative negative Affirmative Negative

1. All squares are No squares are Some squares are Some squares are
similar. similar. similar not similar

All squares are Not all squares are

not similar similar

2. All SHS students No SHS students Some SHS students Some SHS students
are ABM students are ABM are ABM students are not ABM
in this school. students in this in this school. students in this
school. school.

Not all SHS

students are ABM
students in this
school.
3. All animals are No animals are Some animals are Some animals are
found in zoos. found in zoos. found in zoos. not found in zoos.

All animals are

not found in zoos.

Activity 5:
1. q → p
2. ~(q ↔ p)
3. p ∧ q
4. q ∨ ~p
5. ~((q ∧ p) → ~q)

Activity 6:
1. Sir Kesh is a Math teacher or a UP student.
2. Sir Kesh is not a UP student but a Math teacher.
3. If Sir Kesh is a Math teacher, then he is a UP student.
4. Sir Kesh is not a UP student if and only if he is a Math teacher.
5. If Sir Kesh is not a Math teacher or a UP student, then he is a UP student.
6. It is not the case that Sir Kesh is a Math teacher if and only if he is a UP student.

References
Book

Dr. Debbie Marie B. Verzosa, Paolo Luis Apolinario, Regina M. Tresvalles, Francis Nelson
M. Infante, Jose Lorenzo M. Sin, Len Patrick Dominic M. Garces et al., General
Mathematics Teacher’s Guide. Pasig City: Department of Education, First Edition
2016.

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