LEARNING ACTIVITY SHEETS IN

GENERAL MATHEMATICS WEEK 7-8

Learning Competencies:
1. define proposition and illustrate a proposition
2. distinguish between simple and compound propositions.
3. define the logical operators: negation, conjunction, disjunction, conditional, and biconditional
4. perform different types of operations on propositions.

Prepositions
INTRODUCTION :
Math involves logic. There is logic in performing mathematical operations, in playing chess, in making decisions, in
communicating with peers and in many other activities that we do. Actually, we may be using the principles and concepts
of logic in our daily tasks which may be unknowing to us sometimes. That is how the study of logic may be significant to
everybody. To commence our discussion on logic, this lesson will introduce the key concepts of propositions.

Proposition Defined
A Proposition is a declarative sentence that is either true or false, but not both.
A true proposition has a truth value of “true”, otherwise, its truth value is “false”. Oftentimes, a small letter is used to
denote a proposition. That is, the proposition
a: It is getting clearer.
may be read as
a is the proposition “It is getting clearer.”.
Activity 1.
Consider the following sentences. Write P if it is a proposition, and NP if not.
1. a: The largest continent is Asia.
2. b: Do I need to wash my hands regularly?
3. c: Occipital is the part of our brain responsible for vision so we have to be sure that it is taken care of.
4. d: Happy Birthday, Inay Baby!
5. e: The longest bone in the body is the femur.

Simple and Compound Propositions

A Simple Proposition is a proposition that cannot be broken down into more than one proposition. Otherwise, it is a
Compound Proposition. The latter is a proposition that is formed by joining simple propositions using logical
connectors.
Example: Identify each of these as simple or compound proposition.
a: Grounding is beneficial to a person.
d: There is no stronger than the heart of a volunteer.
p: 3! = 6/2
𝑝1: If an individual is great, then there is a teacher behind.
𝑝2: Either a person saves before spending, or one spends before saving.
𝑝3: It is not a shame to greet the utility worker the same way as with the school principal.
𝑝4: If a person is disabled, then he/ she is entitled to obtain a PWD ID, and if a person is entitled to obtain a PWD ID,
then he/ she is disabled.
Solution:
Notice that among the propositions above, propositions a, d and p are made up of one declarative sentence each, while
propositions 𝑝1, 𝑝2, 𝑝3 and 𝑝4 are composed of more than one declarative sentence.
This informs us that propositions a, d and p are all simple propositions, while propositions 𝑝1, 𝑝2, 𝑝3 and 𝑝4 are
compound propositions.

Given propositions p and/or q, some logical connectors

may be expressed in terms of the following:
not p
p and q
p or q
If p, then q

Activity 2
In each number, write SP, CP, or N if it is a simple proposition, compound proposition or not a proposition, respectively.
Furthermore, identify the logical connector/s used if it is a compound proposition.
1. If bad company ruins good morals, then one should be mindful in choosing friends.
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 2. Essential oils do not expire.
3. Ponder on the sacrifices of your parents.
4. If you study hard, then you will have good grades.
5. Have you thought of the sick when you are spending too much?
6. In tossing a fair coin once you may get a head or a tail.
7. There is wisdom in spending for needs and it makes sense to think at least twice before giving in for wants.
8. Two lines are parallel if and only if they are coplanar and do not intersect.
9. Study hard and rest well.
10. Camber is not telling the truth.

Logical Operators
Just like with integers, fractions, rational expressions, polynomials, exponential functions and the many other types of
functions, there are operations involved in propositions.

Logical Operators Enumerated

Negation Defined
The Negation of a proposition p is denoted by ~p which is read as “not p”, and is defined through its truth table

𝑝 ~𝑝
T F
F T
Example 1
State the negation of each of the following propositions.
𝑛1: Quality determines the price.
𝑛2: A learned is one who is educated.
𝑛3: 𝑓(𝑥) = 𝑥2 is a cubic function.
Solution:
~𝑛1: It is not true that quality determines the price or ~𝑛1: Quality does not determine the price.
~𝑛2: A learned is not one who is educated.
~𝑛3: 𝑓(𝑥)= 𝑥2 is not a cubic function or ~𝑛3: It is not true that f(x) = x2 is a cubic function.

Conjunction Defined
Another logical operator is Conjunction of the propositions p and q which is denoted by 𝑝 ∧ 𝑞 and read as “p and q”, and is
defined through its truth table

Proposition p and q are called Conjuncts. The conjunction 𝑝 ∧ 𝑞

is true only when both conjuncts p and q are true.

Let d and e be propositions.

d: Leniency is long-suffering.
e: Those who misunderstand it abuse it.

Express the conjunctions below in verbal sentences or in symbols.

1. 𝑑 ∧ 𝑒
2. ~𝑑 ∧ 𝑒
3. “Leniency is long-suffering and it is not true that those who misunderstand it abuse it.”
4. “Leniency is not long-suffering and those who misunderstand it do not abuse it.”

Answers:
1. Leniency is long-suffering and those who misunderstand it abuse it.
2. Leniency is not long-suffering and those who misunderstand it abuse it.
3. 𝑑 ∧ (~𝑒)
4. ~𝑑 ∧ (~𝑒)

Disjunction Defined
Negation and conjunction do not suffice logic. There are logical statements that connote the disjunction of propositions.
The Disjunction of propositions p and q is denoted by 𝑝 ∨ 𝑞 which read as “𝑝 𝑜𝑟 𝑞”, and defined through its truth table

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 Propositions p and q are each called Disjuncts. The disjunction 𝑝 ∨ 𝑞
is false only when both disjuncts p and q are false.

Let t, u and y be propositions.

t: He is an old soul.
u: Old songs soothe his ears.
y: Old fashion is inviting to his eyes.

Express the disjunctions below in verbal sentences or in symbols.

1. 𝑢 ∨ 𝑦
2. (~𝑡 ^ (~𝑢)) ∨ 𝑦
3. “Either he is an old soul or old songs soothe his ears and old fashion is inviting to his eyes.”
4. “He is not an old soul and either old fashion is not inviting to his eyes or old songs soothe his ears.”

Answers:
1. Old songs soothe his ears or old fashion is inviting to his eyes.
2. Either he is not an old soul and old songs do not soothe his ears or old fashion is inviting to his eyes.
3. 𝑡 ∨ (𝑢 ∧ 𝑦)
4. ~𝑡 ∧ (~𝑦 ∨ 𝑢)

Conditional Defined
Another common kind of logical propositions includes conditional. The Conditional of propositions p and q is denoted by
𝑝 → 𝑞 which read as “if p, then q” or “p implies q”, and defined through its truth table

Proposition p is called a Hypothesis, while proposition q is

called Conclusion.

Let x, y and z be propositions.

x: Dioxins are found almost everywhere.
y: Plastics, bleached paper and most commodities contain the chemical dioxin.
z: Dioxins are culprits to many diseases.
Express the conditionals below in verbal sentences or in symbols, as the case may be.
1. 𝑥 → 𝑧
2. (~𝑦) → (~𝑥 ∧ ~𝑧)
3. “If dioxins are culprits to many diseases, then they are found almost everywhere and plastics, bleached paper and most
commodities contain this chemical.”
4. “If dioxins are not found almost everywhere, then it is not true that either plastics, bleached paper and most commodities
containing this chemical, or dioxins are the culprits to many diseases.”
Answers:
1. If dioxins are found almost everywhere, then they are culprits to many diseases.
2. If plastics, bleached paper and most commodities do not contain the chemical dioxin, then it is not true that dioxins are
found almost everywhere and it is not true that dioxins are a culprit to many diseases.
3. 𝑧 → (𝑥 ∧ 𝑦)
4. (~𝑥) →~(𝑦 ∨ 𝑧)

Biconditional Defined
The last logical proposition that we shall consider is Biconditional. This is denoted by “𝑝 ↔ 𝑞” or “p iff q” given propositions
or Components p and q and it is read as “p if and only if q”, and defined through its truth table

Let g, h, i and j be propositions.

g: Only physically handicapped individuals can be called persons with disabilities.
h: Psychosocially disabled persons like those with chronic illnesses can also avail PWD ID.

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i: There are numerous benefits which come along with owning a PWD ID.
j: Knowledge of the wide extent of who a PWD has yet to be spread.

Express the biconditionals below in verbal sentences or in symbols.

1. (~𝑔 ↔ ℎ) ∧ (𝑗 → 𝑖)
2. (𝑔 ∧ (~ℎ)) ↔ (~𝑗)
3. “Knowledge of the wide extent of who a PWD is has yet to be spread if and only if not only physically handicapped
individuals can be called persons with disability.”

Answers:
1. Not only physically handicapped individuals can be called persons with disability if and only if psychosocially disabled
persons like those with chronic illnesses can also avail PWD ID, and if knowledge on the wide extent of who a PWD has
yet to be spread, then there are numerous benefits which come along with owning a PWD ID.
2. Only physically handicapped individuals can be called persons with disabilities and psychosocially disabled persons like
those with chronic illnesses cannot avail PWD ID if and only if it is not true that knowledge on the wide extent of who a
PWD is has yet to be spread.
3. 𝑗↔ (~𝑔)

Activity 3
Let d, e, f and g be propositions.
d: There is wisdom in spending on needs.
e: It makes sense to think at least twice before giving in for wants.
f: A good planner saves first before spending.
g: A shopaholic should learn self-control.

Express the following propositions in symbols.

1. If a good planner saves first before spending, then there is wisdom in spending on needs and it makes sense to think at
least twice before giving in for wants.
2. A shopaholic should not learn self-control if and only if a good planner does not save first before spending and it does
not make sense to think at least twice before giving in for wants.
3. There is wisdom in spending on needs or a shopaholic should not learn self-control.

For numbers 4, 5and 6, consider the following propositions:

j: A soft answer turns away wrath.
k: A harsh word stirs up anger.
l: A wise man listens before speaking.
4. Which is the symbol for the proposition “A soft answer turns away wrath
and a harsh word stirs up anger”?
a. ~𝑗 ∨ 𝑘
b. 𝑗∧𝑘
c. 𝑗 →𝑘
d. 𝑗↔~𝑘

5. Which is the verbal statement for ~𝑗∧~𝑘?

a. A soft answer does not turn away wrath or a harsh word does not stir up anger.
b. A soft answer does not turn away wrath and a harsh word does not stir up anger.
c. A soft answer turns away wrath and a harsh word stirs up anger.
d. If a soft answer does not turn away wrath, then a harsh word does not stir up anger.

6. Which is the verbal statement for 𝑗→ 𝑘?

a. A soft answer turns away wrath or a harsh word stirs up anger.
b. A soft answer turns away wrath and a harsh word does not stir up anger.
c. If a soft answer turns away wrath, then a harsh word stirs up anger.
d. A soft answer turns away wrath and a harsh word stirs up anger.

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