Simple and
Compound
Proposition
Lesson Objectives
At the end of the lesson the learner should be able to:
Illustrate proposition;
Distinguish between simple and compound
proposition;
Symbolize proposition; and
Perform the different types of operations on
proposition.
CODE: M11GM-IIg-4
The class will be grouped into three. Each group will be given 4
sentences to explain why is it false and make it correct.
1. Do you love math subject? – Is sentence that expresses a strong feeling.
2. Photosynthesis is a process where plants make their food. Is an example
of Exclamatory sentence.
3. Imperative is a sentence that give command. It is ended with a exlamation
point.
4. A compound sentence is a sentence with one independent clause, also
known as the main clause, and one or more dependent clauses.
5. A compound sentence is a structure that contains at least three clauses—
two dependent clauses and one or more independent clauses.
Review Types of Sentences
What are the different kinds of sentences?
Define declarative, imperative, exclamatory
and interrogative sentences.
Can you differentiate simple to compound
sentences?
Engage
Proposition
Proposition is a declarative sentence that is either true or false, but not
both. It is the building block of an argument.
A simple proposition is a short statement and does
not contain any other statement as a part.
A compound proposition are statements more than
two simple statements.
Determine whether each of the following is a
proposition or mere sentence.
1. Do you know the painting Spolarium?
2. “Spoliarium” was the name given to the Roman Colosseum basement
where fallen gladiators were thrown in after combat.
3. Roman cruelty! It is our country mens same grievance under Spanish rule.
4. It illustrates two dead gladiators being dragged by Romans at the center.
5. Look on the left of the painting
6. Can you tell me where can I see Roman that raises a fist in protest?
7. I can see a woman mourns on a loved one on the right side and an old man
searches for a body amid the smoky haze.
8. Spolarium can be seen in national museum National Museum of Fine Arts or
on its online site in its 360-virtual tour.
Can you
the compound
proposition
from sentences?
the given examples?
How
did identify
you identify
propositions
from the given
Engage
Compound propositions are composed of
subpropositions and various connectives or logical
operators. Such composite propositions, just like
compound sentences in languages, may be constructed
using logical connectors which are also called logical
operators.
There are five types of compound proposition and its
logical operator.
Engage
1. Negation – It is any statement P which can be formed by
using the word not. The necation of P in symbols is ~P (read as
not P)
If P represents “1 is an even number,” then its negation (1 is not
an even number” or “1 is an odd number” it is represented by ~P
We write
P: 1 is an even number
~P: 1 is not an even number or 1 is an odd number
Engage
2. Disjuction. If P and Q are propositions, the disjunction of P and Q,
denoted by P ˅ Q which is read as “ P or Q” is the proposition whose
truth value depends on P or Q.
If P represents “2 is a prime number” and E stands for “ 2 is an even
number” then the disjunction “ 2 is a prime or even number” may be
symbolized as P˅E. We write
P: 2 is a prime number
E: 2 is an even number
P˅E: 2 is a prime or even number
Engage
3.Conjunction. If P and Q are propositions, the conjuction of P and Q,
denoted by P ˄ Q which is read as “P and Q” is the proposition whose
truth value depends on P and Q.
If P represents “2 is a prime number” and E stands for 2 is an even number,
“ then the conjunction “2 is a prime and an even number” may be
symbolized as P˄E. We write
P: 2 is a prime number
E: 2 is an even number
P˄E: 2 is a prime and an even number
Engage
4. Implications. These are conditional statements which also
sometimes called as if-then statements. The “if” part is called the
hypothesis or premise and the “then” part is called the conclusion. The
hypothesis and conclusion may be represented by P and Q
respectively and the implication is represented by P→Q.
Engage
4. Implications.
The statement “ If I got perfect score in the examination, then I will
treat you to lunch” is a conditional statement where the hypothesis is
“ I got perfect score in the examination” and the conclusion is “ I will treat
you to lunch”.
We write
P: I got perfect score in the examination
Q: I will treat you to lunch
P→Q: If I got perfect score in the examination, then I will treat you to
lunch.
Engage
For our reference, these are the similar statements for each
logical operators.
Symbols
~P
P˄Q
P˅Q
P→Q
Logical Operators and their Meanings
Translation
Not P, it is not the case that P, it is false that P, it is not true that
P.
P and Q, P moreover Q, P although Q, P still Q, P furthermore
Q, P also Q, P nevertheless Q, P however Q, P yet Q, P but Q.
P or Q, P unless Q
If P then Q, P implies Q, P is a sufficient condition for Q, P only
if Q, Q is a necessary condition for P, Q if P, Q follows from P, Q
provided P, Q whenever P, Q is a logical consequence of P.
Engage
2. Since “neither P nor Q” is the same as “not either P or Q”
and “ both P or Q are not”, then it is denoted by ~(P ˅ Q)
3. The order of the words “both’ and “not” should also be
taken into consideration. These are the representations of the
following phrases:
P and Q are not both ~(P ˄Q)
P and Q are both not ~P ˄ ~Q
Engage
It is also important to determine the proper use of parenthesis, brace or
bracket as grouping marks. Here are the guidelines in using these groupings
symbols.
1. The parenthesis is used whenever the word “both” goes with “and” and
“either’ goes with “or”. We now have the following representations:
both P or Q and R
(P ˅ Q) ˄ R
P or both Q and R
P˅(Q ˄R)
Either P or Q or R
(P˅Q) ˅ R
P and either Q or R
P ˄ (Q ˅ R)
EXPLORE
A. Translate English statements to propositional form
Deepen your
Understanding
G: “ Ginebra wins its first game.”
T: “ Talk and Text wins its first game.”
A: “ Alaska wins its first game.”
1. Alaska and Ginebra wins its first game.
2. Alaska wins its first game only if Talk n’ Text loses its first game.
3. Either Alaska wins its first game and Talk n Text loses its first game or, if Talk n Text wins
its first game, then Ginebra does not win its first game.
4. If Ginebra and Alaska both do not win their first games, then Ginebra and Alaska do not
both win their first game.
EXPLORE
A. Translate English statements to propositional form
G: “ Ginebra wins its first game.”
T: “ Talk and Text wins its first game.”
A: “ Alaska wins its first game.”
Deepen your
Understanding
1.
2.
3.
4.
A˄G
A ˅~T
(A ˄ ~T) ˅ (T →~G)
(~G ˄~A) → ~(G˄A)
1. Alaska and Ginebra wins its first game.
2. Alaska wins its first game only if Talk n’ Text loses its first game.
3. Either Alaska wins its first game and Talk n Text loses its first game or, if Talk n Text wins
its first game, then Ginebra does not win its first game.
4. If Ginebra and Alaska both do not win their first games, then Ginebra and Alaska do not
both win their first game.
EXPLORE
Deepen your
Understanding
B. Translating Propositional forms to English statements.
Let A: “Math is about Problem Solving”
B: “Math is Challenging”
Write the following proposition in plain English.
1. ~B
2. A ˄B
3. A→~B
4. ~(A˅B)
EXPLAIN
Using Capital letter to abbreviate simple propositions, indicate the
letter corresponding to each subpropositions; then, symbolize the
following compound propositions.
1. The word of his mouth were smoother than butter, but war was in his
heart (Psalm 55:21)
2. God hath made man upright; but they have sought out many inventions.
(Ecclesiates 7:29)
3. Promotion cometh neither from the east, nor from the west, nor yet
from the south. (Psalm 75:6)
4. If I didn’t care about you, I wouldn’t get so mad at the things that you
do.
ELABORATE
Generalize our topic for today by filling out the statements
below.
_____1. It is a statement sentence which is either true or false but not both.
_____ 2. A proposition composed of only one propositional variable.
_____ 3. A proposition composed of subpropositions and various connectives
or logical operators.
_____ 4. It is the proposition whose truth value depends on P or Q.
_____ 5. It is the proposition whose truth value depends on P and Q.
_____ 6. Conditional statements which also sometimes called as if-then
statements. The “if” part is called the hypothesis or premise and the “then”
part is called the conclusion.
_____ 7. Symbolize the negation of the proposition.
EVALUATE
Direction: Choose the letter of the best answer. Write the chosen letter on a
separate sheet of paper.
1. p ˅ q is an example of a
a) disjunction
b) negation
c) conditional statement
d) conjunction
2. Let p represent: Brent works this summer.
Let q represent: Brent takes a vacation.
What is the symbolic representation of this statement?
Brent does not work this summer.
a) ∴p
b) ~p
c) ~q
d) ∴~p
3. p: all dogs bark
q: all flowers are yellow
Translate to symbolic notation:
It is not the case that all dogs bark and all flowers are yellow.
a) ~p∧q b) ~p∨~q c) ~p∧~q d) ~(p∧q)
4. Which statement is not an example of a conjunction?
a) I passed Math or I passed Add Math
b) I passed Math but failed Add Maths
c) I passed Math however I failed Add Math
d) I passed Math and failed Add Math
5. Let p, q, and r be the propositions
p: I am a human.
q: I am a robot.
r: I am a cyborg.
a) I am a robot or a human and a cyborg.
b) I am a human and a robot or a cyborg.
c) If am a robot or a human, then I am not a cyborg.
d) If I am a human and a robot, then I am a cyborg.
EXTEND
Study the next lesson about “Determining
the truth Values of Proposition” pg 205