Conjunctions and Disjunctions

Logic

Propositions

OBJECTIVES

At the end of this session, you will be able to:

q
illustrate a proposition;

q
symbolize propositions;

q
distinguish between simple and compound propositions;

q
perform operations on propositions;

q
illustrate truth values of propositions.

Riddle

Who makes it, has no need of it. Who buys it, has no use for it.

Who uses it can’t neither see nor feel it.

What is it?

a coffin

Riddle

What can travel around the world while staying in a corner?

a stamp

Riddle

I'm tall when I'm young and I'm short when I'm old. What am I?

candle

Riddle

What has hands but cannot clap?

clock

Riddle

You can drop me from the tallest building, and I'll be fine, but if you drop me in

water I die. What am I?

paper

Riddle

What has a head and a tail, but no body?

coin

Riddle

Paul's height is six feet, he's an assistant at a butcher's shop, and wears size

9 shoes. What does he weight?

meat

Riddle

There was a green house. Inside the green house there was a white house.

Inside the white house there was a red house. Inside the red house there

were lots of babies. What is it?

watermelon

Riddle

What kind of tree can you carry in your hand?

palm

RECALL

Declarative Sentence – is used to state a fact, wish, intent, or feeling.

example: I wish I could sit beside my crush.

Imperative Sentence – is used to state a command, request, or direction.

example: Please do not answer in chorus.

Interrogative Sentence – is used to ask questions.

example: Did you prepare for the quiz today?

Exclamatory Sentence – is used to express a strong feeling.

example: Keep quite!

Types of Sentences

Proposition

A proposition is a declarative sentence that can be classified as true or

false, but not both.

Proposition

Determine: Proposition or Not a Proposition, then True or False

1.

Philippines is a province of China.

2.

Blink three times.

3.

Clyde is a medical front liner.

4.

All parallelograms are quadrilaterals.

Proposition, False

Not a Proposition

Proposition, False

Proposition, True

Proposition

Kinds of Proposition

Simple Proposition

ü

a declarative sentence subject to affirmation or

denial.

ü conveys one thought with no connecting words.

ü Like a simple sentence, it only has one subject and

one predicate.

Compound Proposition

ü contains two or more simple propositions that are

put together using connective words such as and,

or, if ... then, and if and only if.

Example:

STEM students continues the online learning in Gen

Math.

Example:

STEM students continues the online learning in Gen

Math and NUNS continues the blended learning for the

rest of the academic year.

Proposition

Determine: Simple or Compound

1.

The chair is wooden.

2.

Aristotle and Plato are philosophers.

3.

Her dress is red.

4.

Her dress is blue and her blouse is red.

5.

The board is color green.

Simple

Compound

Simple

Compound

Simple

Simple Proposition

A negation is a simple proposition that denies the truth of the given proposition.

That is, the negation statement is false whenever the given statement is true, and

true whenever the given statement is false.

The negation can be obtained by inserting the word not in the given statement or

by prefixing it with phrases such as “It is not the case that...”

Example:

Proposition: The cat is black.

Negation: The cat is not black.

Proposition

Determine the negation of each proposition.

a. Mei bought the latest smartphone.

b. There is no pollution in the city.

c. 2 + 1 = 3

d. Aga bakes a cupcake.

e. Julius did not ride the bicycle.

Mei did not buy the latest smartphone.

There is pollution in the city.

2 + 1 ≠ 3. We may also say, “It is not the case that 2 + 1 = 3”.

Aga does not bakes a cupcake.

Julius ride the bicycle.

Compound Proposition

In compound propositions, we represent each simple proposition to letters, usually

𝒑, 𝒒, 𝒓, 𝒔, and 𝒕 .

e.g. 𝑝: Adrian is the class president.

𝑞: Faith is the class secretary.

𝑝 and 𝑞

Adrian is the class president and Faith is the class secretary.

Connectives: ∧,∨, ⨁, ⟶, ⟷, ∼

Compound Proposition
Types of Compound Proposition

Connective Symbol

read as
Type of a compound

proposition
In symbols

∧

and

conjunction

𝑝 ∧ 𝑞

∨

or

inclusive disjunction

𝑝 ∨ 𝑞

⨁

or

exclusive disjunction

𝑝 ⊕ 𝑞

⟶
implies
if…then
implication

𝑝 ⟶ 𝑞

⟷

if and only if

equivalence

𝑝 ⟷ 𝑞

~

not

negation

~𝑝

Compound Proposition

Conjunction ∧

e.g. 𝑝: USA has Joe Biden

𝑞: China has Xi Jinping.

𝑝 ∧ 𝑞: USA has Joe Biden AND China has Xi Jinping

Types of Compound Proposition

𝑝

𝑞

𝑝 ∧ 𝑞

T

T

T

T

F

F

F

T

F

F

F

F

Truth Table

Compound Proposition

Disjunction Inclusive ∨

e.g. 𝑝: I will pay my hospital bill in cash

𝑞: I will pay my hospital bill through card.

𝑝 ∨ 𝑞: I will pay my hospital bill in cash or in card.

Types of Compound Proposition

𝑝

𝑞

𝑝 ∨ 𝑞

T

T

T

T

F

T

F

T

T

F

F

F

Truth Table

Compound Proposition

Disjunction Exclusive ⊕

e.g. 𝑝: I am covid-19 positive.

𝑞: I am covid-19 negative.

𝑝⨁𝑞: I am either covid-19 positive or negative.

Types of Compound Proposition

𝑝

𝑞

𝑝⨁𝑞

T

T

F

T

F

T

F

T

T

F

F

F

Truth Table

Compound Proposition

Implication ⟶

An implication is a proposition that claims a given proposition (called

the antecedent) entails another proposition (called the consequent).

Implications are also known as conditional propositions.

e.g. 𝑝: 2 + 5 = 7.

𝑞: 2

5 = 10.

𝑝 ⟶ 𝑞: If 2 + 5 = 7, then 2

5 = 10.

Types of Compound Proposition

𝑝

𝑞

𝑝 ⟶ 𝑞

T

T

T

T

F

F

F

T

T

F

F

T

Truth Table

Compound Proposition

Equivalence ⟷

e.g. 𝑝: Two sides of a triangle are congruent.

𝑞: Two angles opposite them are congruent.

𝑝 ⟷ 𝑞: Two sides of a triangle are congruent if and only if

the two angles opposite them are congruent.

Types of Compound Proposition

𝑝

𝑞

𝑝 ⟷ 𝑞

T

T

T

T

F

F

F

T

F

F

F

T

Truth Table

Compound Proposition

Negation ~

e.g. 𝑝: Manila is the capital city of the Philippines.

~𝑝: Manila is not the capital city of the Philippines.

Operation of Compound Proposition

𝑝

~𝑝

T

F

F

T

Truth Table

Try this

Classify each proposition as simple, compound, or neither. If compound, classify as conjunction, disjunction, conditional,

or biconditional.

a. If it is sunny outside, then I will walk to work.

b. Today is Sunday.

c. If I drive and if it is raining, then I carry my umbrella.

d. Juan is not going to enroll next semester.

e. Every integer is even, or every even natural number is an integer.

a. It is a compound proposition and the connective words
are “if ... then.” Thus, it is a conditional proposition.
b. It is a simple proposition.
c. It is a compound proposition and the connective words
are “and” and “if ...then.” Thus, it is both a conjunction and
a conditional proposition.
d. It is a simple proposition.
e. It is a compound proposition and the connective word is
“or.” Thus, it is a disjunction.

Try this

Given: 𝑝: Xhawn is a Grade 11 student.

𝑞: Xhawn is a STEM student.

Write each symbolism in ordinary English sentence or vice versa.

1. 𝑝 ∧ 𝑞

2. 𝑝 ∨ 𝑞

3. 𝑝 ⟶ 𝑞

4. ~𝑝 ⟷ ~𝑞

Xhawn is a Grade 11 student and a STEM student.

Xhawn is a Grade 11 student or a STEM student.

If Xhawn is a Grade 11 student, then he is a STEM student.

If Xhawn is not a Grade 11 student, then he is not a STEM student.

Try this

Given: 𝑝: Xhawn is a Grade 11 student.

𝑞: Xhawn is a STEM student.

Write each symbolism in ordinary English sentence or vice versa.

1. 𝑝 ∧ 𝑞

2. 𝑝 ∨ 𝑞

3. 𝑝 ⟶ 𝑞

4. ~𝑝 ⟷ ~𝑞

If Xhawn is a STEM student, then he is a Grade 11 student.

If Xhawn is not a STEM student, then he is not a Grade 11 student.

Xhawn is not a G11 student and not a STEM student.

Xhawn is not a STEM student if and only if he is not a Grade 11 student.

Continuation

5. 𝑞 ⟶ 𝑝

6. ~𝑞 ⟶ ~𝑝

7. ~(𝑝 ∧ 𝑞)

8. ~𝑞 ∨ ~𝑝

Recall

As a typical human language has many ways to express the same thought, it is beneficial to study propositions by

translating them into a notation that has a very limited collection of symbols yet is still able to express the basic logic

of the propositions. This statement is called symbolic logic. Mathematical logic is an example of symbolic logic.

Below are terms involving mathematical logic.

A propositional form is an expression involving propositional variables and logical connectives.

Propositional variables are variables that represent propositions, just as letters are used to denote numerical

variables. The conventional letters used for propositional variables are p, q, r, s, and so on.

Compound Proposition
Types of Compound Proposition

Connective Symbol

read as
Type of a compound

proposition

In symbols

(Propositional forms)

∧

and

conjunction

𝑝 ∧ 𝑞

∨

or

inclusive disjunction

𝑝 ∨ 𝑞

⨁

or

exclusive disjunction

𝑝 ⊕ 𝑞

⟶
implies
if…then
implication

𝑝 ⟶ 𝑞

⟷

if and only if

equivalence

𝑝 ⟷ 𝑞

~

not

negation

~𝑝

Try this

Let the following variables represent statements as indicated:

P: The angle sum of a triangle is 180.

Q: 3 + 7 = 10

R: The sine function is continuous.

Then write the following sentences as propositional forms.

a. The sine function is continuous, and 3 + 7 = 10.

b. The angle sum of a triangle is 180°, or the angle sum of a triangle is 180° .

c. If 3 + 7 = 10 , then the sine function is not continuous.

d. The angle sum of a triangle is 180° if and only if the sine function is continuous.

e. The sine function is continuous if and only if 3 + 7 = 10 implies that the angle sum of a triangle is not 180°.

Continuation

Try this

Let p be “Horses have four legs,” q be “17 is a prime number,” and r be “4 × 3 = 12”. Write each of the following

propositional forms in words.

a. ~p ∧ q

b. q → ~r

c. p ∧ (q ∨ r)

d. p ∨ (q ∧ r)

Continuation

Horses do not have four legs and 17 is a prime number.

If 17 is a prime number, then 4 × 3 ≠ 12.

Horses have four legs, and either 17 is a prime number or 4 × 3 = 12.

Either horses have four legs or 17 is a prime number and 4 × 3 = 12.

Try this

1) Let p represent “5 < 8” and q represent “0 > 5.” Find the truth value of p ∧ q.

2) Let p represent “𝑒!= 1” and q represent “0! = 1” Find the truth value of p ∧ q.

Try this

3) Consider the following propositions p, q, and r.

p: Some prime numbers are perfect squares.

q: An equilateral triangle is also equiangular.

r: Zero is neither odd nor even.

Find the truth value of each of the following disjunctions:

a. p ∨ q

b. p ∨ r

c. q ∨ r

d. ~q ∨ p

e. p ∨ ~r

f. ~r ∨ ~p

Try this

4) Consider the propositions p and q below. What value(s) of x will make p ∧ q true?

• p: x is an even number.

• q: x is a prime number

5) Consider the propositions p and q below. For what value(s) of x will the conjunction ~p ∧ q be

false?

• p: x + 3 = 7

• q: 𝑥!= 0

Try this

Determine the truth value of each proposition.

(a). ~ 𝑝 ∧ 𝑞

(b). ~(𝑝 ⟶ ~𝑞)

1. 𝑝 ∧ 𝑞

2. 𝑝 ∨ 𝑞

3. 𝑝 ⟶ 𝑞

4. ~𝑝 ⟷ ~𝑞

Try this

Determine the truth value of each proposition.

(c). 𝑝 ∨ 𝑞 ∧ ~(𝑝 ⨁ 𝑞)

1. 𝑝 ∧ 𝑞

2. 𝑝 ∨ 𝑞

3. 𝑝 ⟶ 𝑞

4. ~𝑝 ⟷ ~𝑞

Try this

Determine the truth value of each proposition.

(d). ( 𝑝 ∧ 𝑞) ∨ 𝑟 ⟶ (~𝑝 ∧ 𝑞)1. 𝑝 ∧ 𝑞

2. 𝑝 ∨ 𝑞

3. 𝑝 ⟶ 𝑞

4. ~𝑝 ⟷ ~𝑞

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