11

General Mathematics Quarter II – Module 8:

Propositional Logic; Syllogisms

and Fallacies

General Mathematics – Grade 11 Self-Learning Module (SLM) Quarter 2 – Module 8: Propositional Logic: Syllogisms and Fallacies First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.

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11

General Mathematics

Quarter II – Module 8:

Propositional Logic; Syllogisms

and Fallacies

Introductory Message

For the facilitator:

Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on

Propositional Logic; Syllogisms and Fallacies!

This module was collaboratively designed, developed and reviewed by educators both

from public and private institutions to assist you, the teacher or facilitator in helping

the learners meet the standards set by the K to 12 Curriculum while overcoming

their personal, social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and independent

learning activities at their own pace and time. Furthermore, this also aims to help

learners acquire the needed 21st century skills while taking into consideration their

needs and circumstances.

In addition to the material in the main text, you will also see this box in the body of

the module:

As a facilitator you are expected to orient the learners on how to use this module.

You also need to keep track of the learners' progress while allowing them to manage

their own learning. Furthermore, you are expected to encourage and assist the

learners as they do the tasks included in the module.

Notes to the Teacher

This contains helpful tips or strategies that

will help you in guiding the learners.

For the learner:

Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on

Propositional Logic; Syllogisms and Fallacies!

The hand is one of the most symbolized part of the human body. It is often used to

depict skill, action and purpose. Through our hands we may learn, create and

accomplish. Hence, the hand in this learning resource signifies that you as a learner

is capable and empowered to successfully achieve the relevant competencies and

skills at your own pace and time. Your academic success lies in your own hands!

This module was designed to provide you with fun and meaningful opportunities for

guided and independent learning at your own pace and time. You will be enabled to

process the contents of the learning resource while being an active learner.

This module has the following parts and corresponding icons:

What I Need to Know

This will give you an idea of the skills or

competencies you are expected to learn in the

module.

What I Know

This part includes an activity that aims to

check what you already know about the

lesson to take. If you get all the answers

correct (100%), you may decide to skip this

module.

What’s In

This is a brief drill or review to help you link

the current lesson with the previous one.

What’s New

In this portion, the new lesson will be

introduced to you in various ways such as a

story, a song, a poem, a problem opener, an

activity or a situation.

What is It

This section provides a brief discussion of the

lesson. This aims to help you discover and

understand new concepts and skills.

What’s More

This comprises activities for independent

practice to solidify your understanding and

skills of the topic. You may check the

answers to the exercises using the Answer

Key at the end of the module.

What I Have Learned

This includes questions or blank

sentence/paragraph to be filled in to process

what you learned from the lesson.

What I Can Do

This section provides an activity which will

help you transfer your new knowledge or skill

into real life situations or concerns.

Assessment

This is a task which aims to evaluate your

level of mastery in achieving the learning

competency.

Additional Activities

In this portion, another activity will be given

to you to enrich your knowledge or skill of the

lesson learned. This also tends retention of

learned concepts.

Answer Key

This contains answers to all activities in the

module.

At the end of this module you will also find:

The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on any part of the

module. Use a separate sheet of paper in answering the exercises.

2. Don’t forget to answer What I Know before moving on to the other activities

included in the module.

3. Read the instruction carefully before doing each task.

4. Observe honesty and integrity in doing the tasks and checking your answers.

5. Finish the task at hand before proceeding to the next.

6. Return this module to your teacher/facilitator once you are through with it.

If you encounter any difficulty in answering the tasks in this module, do not

hesitate to consult your teacher or facilitator. Always bear in mind that you are

not alone.

We hope that through this material, you will experience meaningful learning and

gain deep understanding of the relevant competencies. You can do it!

References This is a list of all sources used in developing

this module.

What I Need to Know

This module was designed and written with you in mind. It is here to help you master

the Propositional Logic; Syllogisms and Falacies. The scope of this module permits it

to be used in many different learning situations. The language used recognizes the

diverse vocabulary level of students. The lessons are arranged to follow the standard

sequence of the course. But the order in which you read them can be changed to

correspond with the textbook you are now using.

This module is comprised of only one lesson:

Truth values of Propositions

Conditional Propositions

After going through this module, you are expected to:

1. determines the truth values of proposition (M11GM-IIh-1);

2. expresses propositions in symbolic form and find its truth value (M11GM-IIh-

1); and

3. illustrate forms of conditional propositions (M11GM-IIh-2);

What I Know

Choose the letter of the best answer. Write the chosen letter on a separate sheet of

paper.

1. It is a declarative sentence that is either true or false, but not both.

A. command

B. imperative

C. proposition

D. paradox

2. It is a declarative sentence which could either be true, nor false, or is both

true and false.

A. command

B. imperative

C. proposition

D. paradox

3. It is a proposition that is composed of at least two simple propositions joined

together by logical connectives.

A. one proposition

B. simple proposition

C. compound proposition

D. uno proposition

4. What is the symbolic form of Negation with P as arbitrary propositions?

A. ~P

B. ≠P

C. -P

D. _P

5. In a conditional proposition, if the hypothesis is false and the conclusion is

true, what is the truth value of the proposition?

A. True

B. False

C. Unknown

D. None of the above

6. Among the different forms of conditional proposition, which of the following

shows the contrapositive form of conditional proposition? Hypothesis: Juan

is a Filipino. Conclusion: He is born in the Philippines.

A. If Juan is a Filipino, then he is born in the Philippines.

B. If he is born in the Philippines, then Juan is a Filipino.

C. If Juan is not a Filipino, then he is not born in the Philippines

D. None of the above

7. If there are only two propositions, p and q, how many rows should be included

in constructing the truth table?

A. 1

B. 2

C. 4

D. 8

8. If p is a proposition, then the following statements are true about ~𝑝, except

A. ~𝑝 is always false.

B. ~𝑝 is always true.

C. ~𝑝 is either true or false.

D. ~𝑝 can never be true of false.

9. What is the meaning of p ∨ q?

A. p ∨ q means p is true

B. p ∨ q means q is true

C. p ∨ q~𝑝 means q or p is true

D. p ∨ q~𝑝 means q or p is true or both are true.

10. State the negation of the proposition “Everyone in Visayas speaks Cebuano.

A. Not everyone in Visayas speaks Cebuano.

B. Everyone in Visayas do not speak Cebuano.

C. Not everyone in Visayas do not speak Cebuano.

D. All of the above.

11. Given the statement p: Jay lives in Sarangani and q: Jay drives a red car, write

the compound proposition “Jay lives in Sarangani and Jay drives a red car”

using the appropriate symbols.

A. p ∧ q

B. p ∨ q

C. 𝑝 → 𝑞

D. 𝑝 ↔ 𝑞

12. Select the statement that is the negation of “You wear matching socks to the

interview or you don’t get hired.”

A. You wear matching socks to the interview and you get hired.

B. You don’t wear matching socks to the interview or you get hired.

C. You don’t wear matching socks to the interview and you don’t get hired.

D. If you don’t wear matching socks to the interview, then you don’t get

hired.

13. Which of the following is the truth table for 𝑝 → ~𝑝.

A.

p q 𝑝 → ~𝑝

T T T

T F F

F T F

F F F

a. (𝑝 ∧ 𝑝) ↔ 𝑝

T

T

T

T

a. (𝑝 ∧ 𝑞) → 𝑝

T

T

T

T

b. (𝑝 ∧ 𝑞) → 𝑝

F

T

T

F

b. (𝑝 ∧ 𝑞) → 𝑝

F

T

T

T

d. (𝑝 ∧ 𝑞) → 𝑝

T

T

T

F

B.

p ~𝑝 𝑝 → ~𝑝

T F F

T F F

F T T

F T T

C.

p q ~𝑝 𝑝 → ~𝑝

T T T T

T F T F

F T F F

F F F F

D.

p q ~𝑞 𝑝 → ~𝑝

T T F F

T F T T

F T F F

F F T T

For numbers 14-15, determine the truth values of the given propositions.

14. (𝑝 ∧ 𝑝) ↔ 𝑝

15. (𝑝 ∧ 𝑞) → 𝑝

b. (𝑝 ∧ 𝑝) ↔ 𝑝

F

F

F

F

c. (𝑝 ∧ 𝑝) ↔ 𝑝

T

F

T

T

d. (𝑝 ∧ 𝑝) ↔ 𝑝

F

T

T

T

Lesson

1 Truth Values of Propositions

Recall that a simple proposition is a statement with truth values, that is, the

statement is either true or false. In our previous lesson, we performed the different

operations of compound propositions, namely: conjunction, disjunction, implication,

equivalence, and negation. In constructing truth tables and determining the truth

values of a proposition, mastering these five operations is an essential skill.

What’s In

In the previous lesson, we discussed about the simple and compound

propositions and the types of compound propositions.

Type of

Compound

Proposition

Logical Operator 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

By now, you should have mastered the truth table of the logical operators

because we are going to use them to determine all the possible truth values of more

complex compound propositions using truth tables.

What’s New

Constructing Truth Tables

Combining propositions using connective will result in the formation of the new

compound proposition. To determine whether this resulting proposition is true or

false, a scheme for listing all possible truth values will be used. This is called the

truth table.

Activity 1. “Complete Me”

Complete the truth tables below.

What is It

Example 1. Let p and q be propositions. Construct the truth table for the proposition

~(𝒑 ∧ 𝒒).

Solution. Note that there are two propositions, p and q, involved in the compound

proposition. Thus, the truth table will contain 4 rows. First, will construct the

proposition 𝒑 ∧ 𝒒 from ~(𝒑 ∧ 𝒒) and that is:

p q 𝑝 ∧ 𝑞

T T T

T F F

F T F

F F F

Then we will add another column to the right of the proposition ~(𝒑 ∧ 𝒒), the negation

of 𝐭𝐡𝐞 𝐭𝐡𝐢𝐫𝐝 𝐜𝐨𝐥𝐮𝐦𝐧:

p q 𝑝 ∧ 𝑞 ~(𝑝 ∧ 𝑞)

T T T F

T F F T

F T F T

F F F T

1.

p q 𝑝 ∧ 𝑞

T T

T F

F T

F F

2.

p q 𝑝 → 𝑞

T T

T F

F T

F F

3.

p q 𝑝 ↔ 𝑞

T T

T F

F T

F F

4.

p q 𝑝 ∨ 𝑞

T T

T F

F T

F F

Example 2. Let p and q be propositions. Construct the truth table for the compound

proposition (𝒑 → 𝒒) ∧ (𝒒 → 𝒑).

Solution. Note that there are two propositions, p and q, involved in the compound

proposition. Thus, the truth table will contain 4 rows. Using the truth table for the

definition of the conditional statements 𝑝 → 𝑞 and 𝑞 → 𝑝, we add two more columns

to indicate the truth values of 𝑝 → 𝑞 and 𝑞 → 𝑝:

p q 𝑝 → 𝑞 𝑞 → 𝑝

T T T T

T F F T

F T T F

F F T T

In the final column, we add the truth value of (𝑝 → 𝑞) ∧ (𝑞 → 𝑝), which is a conjunction

of the third and the fourth column.

p q 𝑝 → 𝑞 𝑝 → 𝑞 (𝑝 → 𝑞) ∧ (𝑞 → 𝑝)

T T T T T

T F F T F

F T T F F

F F T T T

Example 3. Determine the truth value of the compound proposition ~(𝑝 ∧ 𝑞) ∨ ~(𝑞 ↔ 𝑝).

Solution. The given proposition is a disjunction of two to compound proposition. So,

we need to construct a table with 4 rows and 4 columns and find the truth values.

We have

p q 𝑝 ∧ 𝑞 𝑞 ↔ 𝑝

T T T T

T F F F

F T F F

F F F T

We will add another two columns for the negation of the 3rd and 4th columns. To find

their values negate the 3rd and 4th columns respectively. that is:

p q 𝑝 ∧ 𝑞 𝑞 ↔ 𝑝 ~(𝑝 ∧ 𝑞) ~(𝑞 ↔ 𝑝)

T T T T F F

T F F F T T

F T F F T T

F F F T T F

Finally, in our last step, we will add another column for the final proposition. The

truth values of this column are the disjunction of the 5th and 6th columns.

p q 𝑝 ∧ 𝑞 𝑞 ↔ 𝑝 ~(𝑝 ∧ 𝑞) ~(𝑞 ↔ 𝑝) ~(𝑝 ∧ 𝑞) ∨ ~(𝑞 ↔ 𝑝)

T T T T F F F

T F F F T T T

F T F F T T T

F F F T T F T

Lesson

2 Conditional Propositions

In your daily lives, you are faced with numerous issues you need to resolve

and decide on. How do you decide? Do you base on your emotions or on logic? Or do

you consider other factors as well? Do you think of some conditions that may affect

your decision?

What’s In

In the previous lesson, we discussed about the simple and compound

propositions and the types of compound propositions. Your knowledge on their

symbols, logical operators and truth values are important for you to ace this lesson!

Make sure you familiarize them for you will encounter them in this part of the

modules as well.

What’s New

To help you understand Conditional Propositions, think of the situation below:

Your friend promises you that if he/she save some money, he/she will buy a

gift for you. Below is your friend’s statement.

“If I save some money, then I will buy a gift for you.”

How will you believe that your friend’s promise is true?

Scene A. Your friend saves some money then buys you a gift.

Question: Is your friend’s promise is true?

Answer: Yes, because your friends save some money and buys you a gift.

Scene B. Your friend does not save any money, but still buys you a gift.

Question: Is your friends promise is true?

Answer: Yes, because even your friend does not save any money he still buys

you a gift.

Scene C. Your friend saves some money, but did not buy you a gift.

Question: Is your friends promise is true?

Answer: No, because even your friend saves some money he did not buy you

a gift.

Scene D. Your friend does not save some money and did not buy you a gift.

Question: Is your friends promise is true?

Answer: Yes, because your friend does not save some money that’s why he

did not buy you a gift.

What is It

Conditional Proposition

The following example illustrates that one must be careful in interpreting

conditionals.

Example 1: Suppose that Geebee is a Grade 11 Student. Consider the

following conditionals:

p1: If Geebee is in Grade 11, then she is a senior high school student.

p2: If Geebee is in Grade 11, then she is working as a lawyer.

Definition.

The Conditional of propositions p and q is denoted by

p→ q: (If p, then q.)

and is defined through its truth table:

p q p→ q

T T T

T F F

F T T

F F T

The conditional p→ q may also be read as ‘p implies q’. The proposition p

is called the hypothesis, while the proposition q is called the conclusion.

p3: If Geebee has a degree in Computer Science, then she believes in true love.

Solution.

p1: 'Geebee is in Grade 11' and 'Geebee is a senior high school student' are

both true. Since the hypothesis and conclusion are both true, then p1 is

true, as the first row of the truth table asserts.

p2: 'Geebee is in Grade 11' is true and 'Geebee is working as a lawyer' is false

because a Grade 11student is not yet qualified to be a lawyer. Since the

hypothesis is true but the conclusion is false, then p2 is false, as the

second row of the truth table indicates.

p3: 'Geebee has a degree in Computer Science' is false because Geebee is still

in Grade 11, and so cannot not possibly have a university degree yet. On

the other hand, we do not know the truth value of 'Geebee believes in true

love'. However, according to the last two rows of the truth table, p3 is true

regardless of the truth value of its conclusion.

Another way to understand the truth value of conditional proposition p → q is

to think of it as a promise or a contract. The conditional p → q is false or, equivalently,

the promise is broken when the hypothesis p is true, while the conclusion q is false.

Example 2.

One day, Richard tweeted to the utter delight of his friends: 'If I get promoted,

then I will stop posting selfies on Facebook.' Let p be the statement 'Richard gets

promoted' and let q be the statement, 'Richard stops posting selfies on Facebook.'

Determine whether the conditional p → q is true in each of the following scenarios.

Scenario A: Richard got the promotion and promptly stopped posting selfies

on Facebook.

Scenario B: Richard got promoted, but then he realized that posting selfies

has become a habit that he cannot break easily.

Scenario C: Richard stopped posting selfies, but he did not get the promotion.

Solution.

Scenario A: In this scenario both the hypothesis and the conclusion are true.

This means that Richard kept his promise. According to the first row of the truth

table, the conditional p → q is true.

Scenario B: In this scenario, Richard broke his promise. According to the

second row of the truth table, the conditional p → q is false.

Scenario C: In this scenario, did Richard break his promise? Of course not.

He did not say that his promotion is the only way to make him stop posting selfies.

Perhaps Richard read an article which claims that people who are fond of posting

selfies have some psychological disorder! In any case he did not break his promise,

and according to the third and fourth rows of the truth table the conditional p → q

is true.

In fact, the truth table asserts that when the hypothesis p is false, the

conditional p → q is true regardless of the truth value of the conclusion q. One way

to look at this is that there is no way that Richard can break his promise, when he

fails to get the promotion.

Forms of Conditional Propositions

Conditional Propositions have different forms: Converse, inverse and

contrapositive. They are described respectively below:

Example 1.

Given the conditional proposition below, state its converse, inverse and

contrapositive in a sentence form.

“If you are honest, then you deserve a thumbs-up.”

Solution.

First, identify the hypothesis and the conclusion. Since the proposition is

written in its standard form, it is easily seen that the hypothesis and the

conclusion are as follows:

Hypothesis: You are honest.

Conclusion: You deserve a thumbs-up.

Following the form of converse, inverse and contrapositive in the given

definition, you will have the following:

Converse: If you deserve a thumbs-up, then you are honest.

Inverse: If you are not honest, then you do not deserve a thumbs-up.

Contrapositive: If you do not deserve a thumbs-up, then you are not honest.

Converse: q → p Inverse: ~p → ~ q Contrapositive: ~q → ~p

What’s More

Activity 1. “Tell Me the Truth”

1. Determine the truth value of the proposition ~(𝑝 ∨ 𝑞) by completing the table below.

p q 𝑝 ∨ 𝑞 ~(𝑝 ∨ 𝑞)

T T

T F

F T

F F

2. Determine the truth value of the proposition (𝑝 ∧ 𝑞) → ~(𝑞 ↔ 𝑝) by completing

the table below. p q 𝑝 ∧ 𝑞 𝑝 ↔ 𝑞 (𝑝 ∧ 𝑞) → ~(𝑞 ↔ 𝑝)

T T

T F

F T

F F

Activity 2

Given the conditional proposition identify the hypothesis and conclusion

then state its converse, inverse and contrapositive in a sentence form.

1. If you live in Philippines, then you live in Sarangani Province.

Hypothesis: ________________________

Conclusion: _________________________

Converse:________________________________________________________

Inverse: _________________________________________________________

Contrapositive: __________________________________________________

2. If x is a positive real number, then x2 + 8x is a positive real number.

Hypothesis: ________________________

Conclusion: _________________________

Converse:________________________________________________________

Inverse: _________________________________________________________

Contrapositive: _________________________________________________

What I Have Learned

Fill in the blanks.

1. A statement with truth values, either true or false but not both, is called a

_________________.

2. The conditional proposition p and q is denoted by ______ read as ‘If p, then q’.

3. Conditional Proposition is composed with two statements, the ___________ and

the _____________.

4. Lastly, a conditional proposition comes in 3 forms, namely, ______________, ________________ and ________________.

What I Can Do

Activity 1.

Express the proposition in symbolic form and construct a truth table below.

It is not the case that if the crown prince died or he will marry a commoner,

his younger sister would inherit the throne.

Activity 2.

For this activity, think of an “if” situation in your life right know. Write in

conditional form and identify the hypothesis and conclusion. Lastly, give the

converse, inverse and contrapositive of the statement.

Sample conditional statement: If I will be good in school, then I will finish

Senior High School on time.

In this part below, write your answers:

Conditional Statement 1:

________________________________________________________________________

Hypothesis: ________________________________________________________

Conclusion: ________________________________________________________

Converse: _____________________________________________________________

Inverse: _______________________________________________________________

Contrapositive: ________________________________________________________

Assessment

Choose the letter of the best answer. Write the chosen letter on a separate sheet of

paper.

1. It is a declarative sentence that is either true or false, but not both.

A. command

B. imperative

C. proposition

D. paradox

2. It is a declarative sentence which could either be true, or false, or is

both true and false.

a. command

b. imperative

c. proposition

d. paradox

3. It is a proposition that is composed of at least two simple propositions joined

together by logical connectives.

a. one proposition

b. simple proposition

c. compound proposition

d. uno proposition

4. What is the symbolic form of Negation with P as arbitrary propositions?

a. ~P

b. ≠P

c. -P

d. _P

5. In a conditional proposition, if the hypothesis is false and the conclusion is

true, what is the truth value of the proposition?

a. True

b. False

c. Unknown

d. None of the above

6. Among the different forms of conditional proposition, which of the following

shows the contrapositive form of conditional proposition? Hypothesis: Juan

is a Filipino. Conclusion: He is born in the Philippines.

a. If Juan is a Filipino, then he is born in the Philippines.

b. If he is born in the Philippines, then Juan is a Filipino.

c. If Juan is not a Filipino, then he is not born in the Philippines

d. None of the above

7. If there are only two propositions, p and q, how many rows should be included

in constructing the truth table?

a. 1

b. 2

c. 4

d. 8

8. If p is a proposition, then the following statements are true about ~𝑝, except

a. ~𝑝 is always false.

b. ~𝑝 is always true.

c. ~𝑝 is either true or false.

d. ~𝑝 can never be true or false.

9. What is the meaning of p ∨ q?

a. p ∨ q means p is true

b. p ∨ q means q is true

c. p ∨ q~𝑝 means q or p is true

d. p ∨ q~𝑝 means q or p is true or both are true.

10. State the negation of the proposition “Everyone in Visayas speaks Cebuano.

a. Not everyone in Visayas speaks Cebuano.

b. Everyone in Visayas do not speak Cebuano.

c. Not everyone in Visayas do not speak Cebuano.

d. All of the above.

11. Given the statement p: Jay lives in Sarangani and q: Jay drives a red car, write

the compound proposition “Jay lives in Sarangani and Jay drives a red car”

using the appropriate symbols.

a. p ∧ q

b. p ∨ q

c. 𝑝 → 𝑞

d. 𝑝 ↔ 𝑞

12. Select the statement that is the negation of “You wear matching socks to the

interview or you don’t get hired.”

a. You wear matching socks to the interview and you get hired.

b. You don’t wear matching socks to the interview or you get hired.

c. You don’t wear matching socks to the interview and you don’t get hired.

d. If you don’t wear matching socks to the interview, then you don’t get

hired.

13. Which of the following is the truth table for 𝑝 → ~𝑝.

a.

p q 𝑝 → ~𝑝

T T T

T F F

F T F

F F F

a. (𝑝 ∧ 𝑝) ↔ 𝑝

T

T

T

T

a. (𝑝 ∧ 𝑞) → 𝑝

T

T

T

T

b. (𝑝 ∧ 𝑞) → 𝑝

F

T

T

F

b. (𝑝 ∧ 𝑞) → 𝑝

F

T

T

T

d. (𝑝 ∧ 𝑞) → 𝑝

T

T

T

F

b.

p ~𝑝 𝑝 → ~𝑝

T F F

T F F

F T T

F T T

c.

p q ~𝑝 𝑝 → ~𝑝

T T T T

T F T F

F T F F

F F F F

d.

p q ~𝑞 𝑝 → ~𝑝

T T F F

T F T T

F T F F

F F T T

For numbers 14-15, determine the truth values of the given propositions.

14. (𝑝 ∧ 𝑝) ↔ 𝑝

15. (𝑝 ∧ 𝑞) → 𝑝

b. (𝑝 ∧ 𝑝) ↔ 𝑝

F

F

F

F

c. (𝑝 ∧ 𝑝) ↔ 𝑝

T

F

T

T

d. (𝑝 ∧ 𝑝) ↔ 𝑝

F

T

T

T

Additional Activities

Activity 1.

Determine the truth value of each proposition by constructing its truth tables.

1. ~(~𝑝 → 𝑞)

2. ~(𝑝 ↔ 𝑞)

3. ~(𝑝 ∧ 𝑞) ↔ (𝑝 ∨ 𝑞)

Activity 2.

Given the hypothesis and the conclusion below, write its conditional proposition

form, converse, inverse and contrapositive. (2 points each)

Hypothesis: I eat vegetables. Conclusion: I am a vegetarian. Conditional proposition: _____________________________________________________________________ Converse: _____________________________________________________________________ Inverse: _____________________________________________________________________ Contrapositive: _____________________________________________________________________

Answer Key

What I

Know/Assessment

1.C 2.D 3.C 4.A 5.A 6.D 7.C 8.C 9.D 10.A 11.A 12.C 13.B 14.A 15.A

Lesson 1- What’s New

1.

p q 𝑝 ∧ 𝑞

T T T

T F F

F T F

F F F

2.

p q 𝑝→ 𝑞

T T T

T F F

F T T

F F T

3.

p q 𝑝↔ 𝑞

T T T

T F F

F T F

F F T

4.

p q 𝑝∨ 𝑞

T T T

T F T

F T T

F F F

What I Have Learned

1.proposition

2.𝑝→ 𝑞 3.hypothesis, conclusion 4.converse, inverse,

contrapositive

What's More

Activity 1

1.

p q 𝑝 ∨ 𝑞 ~(𝑝 ∨ 𝑞)

T T T F

T F T F

F T T F

F F F T

2.

p q 𝑝∧𝑞 𝑝↔𝑞 (𝑝 ∧ 𝑞)→~(𝑞↔𝑝)

T T T T T

T F F F T

F T F F T

F F F T T

Activity 2

1.If you live in Philippines, then you live in Sarangani Province.

Hypothesis: You live in Philippines

Conclusion: You live in Sarangani Province

Converse: If you live in Sarangani Province, then you live in the Philippines.

Inverse: If you do not live in the Philippines, then you do not live in sarangani

Province.

Contrapositive: If you do not live in Sarangani Province, then you do not live in

the Philippines.

English.

2.If x is a positive real number, then x2 + 8x is a positive real number.

Hypothesis: x is a positive real number

Conclusion: x2 + 8x is a positive real number

Converse: If x2 + 8x is a positive real number, then x is a positive real number.

Inverse: If x is not a positive real number, then x2 + 8x is not a positive real

number.

Contrapositive: If x2 + 8x is a positive real number, then x is a positive real

number.

What I Can Do

Activity 1

1.Let p: The crown prince died. q: The crown prince will marry a commoner. r: His younger sister would inherit the throne. “The crown prince died or he will marry a commoner”

p ∨ q “If the crown prince died or he will marry a commoner, his younger sister will inherit the throne”

(p ∨ q) →𝒔 “It is not true that if the crown prince died or he will marry a commoner, his younger sister will inherit the throne”

~[(p ∨ q) →𝒔]

p q s p ∨ q (p ∨ q) →𝑠 ~[(p ∨ q) →𝑠]

T T T T T F

T T F T F T

T F T T T F

T F F T F T

F T T T T F

F T F T F T

F F T F T F

F F F F T F

Additional Activities

Activity 1

1.p ∨ ~q

p q ~q 𝑝 ∧ ~q

T T F T

T F T T

F T F T

F F T T

2.~(~𝑝∧𝑞)

p q ~p ~𝑝∧𝑞 ~(~𝑝∧𝑞)

T T F F T

T F F F T

F T T F T

F F T F T

3.~(~𝑝→𝑞)

p q ~p ~𝑝→𝑞 ~(~𝑝→𝑞)

T T F T F

T F F T F

F T T F T

F F T F T

Activity 2

Conditional Proposition: If I eat vegetables, then I am a vegetarian.

Converse: If I am a vegetarian, then I eat vegetables.

Inverse: If I don’t eat vegetables, then I am not a vegetarian.

Contrapositive: If I am not a vegetarian, then I don’t eat vegetables.

References

General Mathematics Published 2016 by C&E Publishing Inc. pp 170-176

Origens, Fernando. Next Century Mathematics: General Mathematics (Quezon City:

Phoenix Publishing House, Inc., 2016), 278-287.

Teachers’ Guide for General Mathematics.pdf pp 249-268 & 261-264

For inquiries or feedback, please write or call: Department of Education – SOCCSKSARGEN Learning Resource Management System (LRMS)

Regional Center, Brgy. Carpenter Hill, City of Koronadal

Telefax No.: (083) 2288825/ (083) 2281893

Email Address: region12@deped.gov.ph