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
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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:
Notes to the Teacher
This contains helpful tips or strategies that
will help you in guiding the learners.
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.
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:
References
This is a list of all sources used in developing
this module.
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!
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:
determines the truth values of proposition (M11GM-IIh-1);
2. expresses propositions in symbolic form and find its truth value (M11GM-IIh1); and
3. illustrate forms of conditional propositions (M11GM-IIh-2);
1.
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
B.
p
T
T
F
F
~𝑝
F
F
T
T
𝑝 → ~𝑝
F
F
T
T
C.
p
T
T
F
F
q
T
F
T
F
~𝑝
T
T
F
F
𝑝 → ~𝑝
T
F
F
F
p
T
T
F
F
q
T
F
T
F
~𝑞
F
T
F
T
𝑝 → ~𝑝
F
T
F
T
D.
For numbers 14-15, determine the truth values of the given propositions.
14. (𝑝 ∧ 𝑝) ↔ 𝑝
a.
(𝑝 ∧ 𝑝) ↔ 𝑝
T
T
T
T
15. (𝑝 ∧ 𝑞) → 𝑝
b.
(𝑝 ∧ 𝑝) ↔ 𝑝
F
F
F
F
c.
d.
(𝑝 ∧ 𝑝) ↔ 𝑝
T
F
T
T
(𝑝 ∧ 𝑝) ↔ 𝑝
F
T
T
T
a.
b.
b.
d.
(𝑝 ∧ 𝑞) → 𝑝
T
T
T
T
(𝑝 ∧ 𝑞) → 𝑝
F
T
T
F
(𝑝 ∧ 𝑞) → 𝑝
F
T
T
T
(𝑝 ∧ 𝑞) → 𝑝
T
T
T
F
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.
1.
2.
p
T
T
F
F
q
T
F
T
F
𝑝∧𝑞
3.
p
T
T
F
F
q
T
F
T
F
𝑝→ 𝑞
p
T
T
F
F
q
T
F
T
F
𝑝∨ 𝑞
4.
p
T
T
F
F
q
T
F
T
F
𝑝↔ 𝑞
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
T
T
F
F
q
T
F
T
F
𝑝∧𝑞
T
F
F
F
Then we will add another column to the right of the proposition ~(𝒑 ∧ 𝒒), the negation
of 𝐭𝐡𝐞 𝐭𝐡𝐢𝐫𝐝 𝐜𝐨𝐥𝐮𝐦𝐧:
p
T
T
F
F
q
T
F
T
F
𝑝∧𝑞
T
F
F
F
~(𝑝 ∧ 𝑞)
F
T
T
T
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
T
T
F
F
q
T
F
T
F
𝑝→𝑞
T
F
T
T
𝑞→𝑝
T
T
F
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
T
T
F
F
q
T
F
T
F
𝑝→𝑞
T
F
T
T
𝑝→𝑞
T
T
F
T
(𝑝 → 𝑞) ∧ (𝑞 → 𝑝)
T
F
F
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
~(𝑞 ↔ 𝑝)
𝑝∧𝑞
𝑞↔𝑝
~(𝑝 ∧ 𝑞)
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
T
T
F
F
q
T
F
T
F
𝑝∧𝑞
T
F
F
F
𝑞↔𝑝
T
F
F
T
~(𝑝 ∧ 𝑞)
F
T
T
T
~(𝑞 ↔ 𝑝)
F
T
T
F
~(𝑝 ∧ 𝑞) ∨ ~(𝑞 ↔ 𝑝)
F
T
T
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
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
T
T
F
F
q
T
F
T
F
p→ q
T
F
T
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.
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.
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:
Converse: q → p
Inverse: ~p → ~ q
Contrapositive: ~q → ~p
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.
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
T
T
F
F
q
𝑝∧𝑞
𝑝↔𝑞
(𝑝 ∧ 𝑞) → ~(𝑞 ↔ 𝑝)
T
F
T
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.
B.
C.
D.
command
imperative
proposition
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
a.
b.
c.
d.
is the symbolic form of Negation with P as arbitrary propositions?
~P
≠P
-P
_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
b.
p
T
T
F
F
~𝑝
F
F
T
T
𝑝 → ~𝑝
F
F
T
T
c.
p
T
T
F
F
q
T
F
T
F
~𝑝
T
T
F
F
𝑝 → ~𝑝
T
F
F
F
p
T
T
F
F
q
T
F
T
F
~𝑞
F
T
F
T
𝑝 → ~𝑝
F
T
F
T
d.
For numbers 14-15, determine the truth values of the given propositions.
14. (𝑝 ∧ 𝑝) ↔ 𝑝
a.
(𝑝 ∧ 𝑝) ↔ 𝑝
T
T
T
T
15. (𝑝 ∧ 𝑞) → 𝑝
b.
(𝑝 ∧ 𝑝) ↔ 𝑝
F
F
F
F
c.
d.
(𝑝 ∧ 𝑝) ↔ 𝑝
T
F
T
T
(𝑝 ∧ 𝑝) ↔ 𝑝
F
T
T
T
a.
b.
b.
d.
(𝑝 ∧ 𝑞) → 𝑝
T
T
T
T
(𝑝 ∧ 𝑞) → 𝑝
F
T
T
F
(𝑝 ∧ 𝑞) → 𝑝
F
T
T
T
(𝑝 ∧ 𝑞) → 𝑝
T
T
T
F
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:
_____________________________________________________________________
Lesson 1- What’s New
1.
p
T
T
F
F
q
T
F
T
F
𝑝∧𝑞
T
F
F
F
What I Have Learned
1.
2.
3.
4.
proposition
𝑝→ 𝑞
hypothesis, conclusion
converse, inverse,
contrapositive
2.
p
T
T
F
F
q
T
F
T
F
𝑝→ 𝑞
T
F
T
T
3.
p
T
T
F
F
q
T
F
T
F
𝑝↔ 𝑞
T
F
F
T
4.
p
T
T
F
F
q
T
F
T
F
𝑝∨ 𝑞
T
T
T
F
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
Answer Key
What's More
Activity 1
1.
p
T
T
F
F
q
T
F
T
F
𝑝∨𝑞
T
T
T
F
~(𝑝 ∨ 𝑞)
F
F
F
T
2.
p
T
T
F
F
q
T
F
T
F
𝑝∧𝑞
T
F
F
F
𝑝↔𝑞
T
F
F
T
(𝑝 ∧ 𝑞) → ~(𝑞 ↔ 𝑝)
T
T
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
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
s
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
T
T
T
T
T
T
T
F
F
(p ∨ q) → 𝑠
p∨q
~[(p ∨ q) → 𝑠]
F
T
F
T
F
T
F
F
Additional Activities
Activity 1
1. p ∨ ~q
p
T
T
F
F
q
T
F
T
F
~q
F
T
F
T
𝑝 ∧ ~q
T
T
T
T
2. ~(~𝑝 ∧ 𝑞)
p
T
T
F
F
q
T
F
T
F
~p
F
F
T
T
~𝑝 ∧ 𝑞
F
F
F
F
~(~𝑝 ∧ 𝑞)
T
T
T
T
3. ~(~𝑝 → 𝑞)
p
T
T
F
F
q
T
F
T
F
~p
F
F
T
T
~𝑝 → 𝑞
T
T
F
F
~(~𝑝 → 𝑞)
F
F
T
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