LOGIC in general mathematics123456789011

Activity: Intelligence Test
The following short IQ test consists of 4 short
questions which test your intelligence, and the
results will tell you, whether you are truly a
manager/leader or a child. The questions like:
"How do you put a giraffe into a refrigerator?"
are easy — the answers may be not: The
questions are NOT that difficult. Get a piece of
paper and answer the following questions
consecutively.

How do you put a
giraffe into a
refrigerator?

How do you put
an elephant into a
refrigerator?

The Lion King is hosting
an animal conference. All
the animals attend except
one. Which animal did not
attend?

There is a river you must
cross but it is used by
crocodiles, and you do not
have a boat. How do you
manage it?

LOGIC
What is a proposition?
Types of Proposition
Operations on Proposition
Truth Values and Truth Tables
Forms of Conditional Proposition
Validity of an Argument

What is a proposition?
Proposition is a complete
declarative sentence that is
either true or false, but not
both.

What is a proposition?
Examples:
1. Manila is the capital of the Philippines.
2. La Union is in Pangasinan.
3.
4.

Types of proposition
Simple proposition is a complete
sentence that conveys one thought with
no connecting words.
Examples:
1. Five is a counting number.
2. Today is Monday.

Compound proposition is a
proposition that is being built up by
combining propositions using
propositional connectives.
Examples:
1. Bonnie is early and Clyde was late.
2. Either he took my coat or someone stole it.

Propositional connective is an
operation that combines two
propositions and to yield a new
proposition whose truth value depends
only on the truth values of the two
original propositions.

The following are the propositional
connectives used in constructing compound
propositions:
Symbol
Conjunction and
Disjunction or
Conditional if
Biconditional if and only if
Negation not

Operations on proposition
Conjunction ()
It is a connection of two simple statement using
the word and or but.
Examples:
1. Bonnie is early and Clyde was late.
2. Bonnie is early but Clyde was late.

Disjunction ()
the word or.
Examples:
1. Either he took my coat or someone stole it.
2. I will pass history or I will be sad.

 Inclusive Disjunction ()
the word or but in the inclusive sense of and/or.
Examples:
1. The weather forecast calls for rain or snow.
2. I will pass history or I will be happy.

 Exclusive Disjunction ()
the word or but in the exclusive sense of either but not
both.
Examples:
1. I will get an A or B for this course.
2. I will pass history or I will fail.

Conditional ()
the word if and then.
Examples:
1. If you will get high scores, then you will pass.
2. If you will pass the exam, then you will get a high grade.
antecedent
antecedent
consequence
consequence

Biconditional ()
the word if and only if.
Examples:
1. You will get high scores if and only if you will review.
2. You will pass if and only if you will get high scores.

Negation ()
the word not.
Examples:
1. Today is not Monday.
2. I will not pass the subject.

Let p = Today is Monday. q = I am tired.
Direction: Transform the following into symbols.
STATEMENT SYMBOL
1. Today is not Monday.
2. Today is Monday and I am tired.
3. Today is Monday and I am not tired.
4. Today is not Monday and I am tired.
5. Today is not Monday and I am not tired.
6. Today is Monday or I am tired.
7. Today is not Monday or I am tired.
∼ 𝒑
𝒑 ∧ ∼𝒒
𝒑 ∧ 𝒒
∼ 𝒑∧ 𝒒
𝒑 ∨ 𝒒
∼ 𝒑∧∼𝒒
∼ 𝒑∨ 𝒒

Let p = I will go swimming.
q = I will go cycling.
r = I will go to the movies.
Direction: Transform the following into symbols.
STATEMENT SYMBOL
1. I will go swimming or I will go cycling, and
I will go to the movies.
2. I will go swimming, or I will go cycling and
I will go to the movies.
(𝒑 ∨𝒒)∧𝒓
𝒑 ∨(𝒒∧𝒓 )

DOMINANT ORDER OF
CONNECTIVES
1. Biconditional
2. Conditional
3. Conjunction/Disjunction
4. Negation
 A symbol outside the parentheses dominates or
outranks any symbol inside the parentheses.

Truth values and truth tables
 The truth table gives us the truth value of a compound
proposition for each possible combination of the truth
or falsity of the simple proposition within the
compound proposition.
 If there are n simple proposition in a compound
proposition, then there are possible true – false
combinations.
 If a proposition is true, its truth value is true, denoted
by T.
 If a proposition is false, its truth value is false,
denoted by F.

T T T T F T T
T F F T T F F
F T F T T T F
F F F F F T T
T F
F T

Direction: Perform the indicated operations using a truth table.
1. 2.

 A compound proposition that is always true, no matter
what the truth values of the propositions that occur in
it is called a tautology.
 A compound proposition that is always false is called a
contradiction.
 A compound proposition that is neither a tautology
nor a contradiction is called a contingency.

Direction: Perform the operations and determine whether the
propositions are tautology, contradiction or contingency.
1. 2.
3.

Logical equivalences
LOGICAL EQUIVALENCE NAME
Identity Laws
Domination Laws
Idempotent Laws
Double Negation Law
Commutative Laws
Associative Laws
Distributive Laws
De Morgan’s Laws

FORMS OF
CONDITIONAL
PROPOSITION

Forms of conditional proposition
STATEMENT SYMBOLS
Conditional
Converse
Inverse
Contrapositive

Forms of conditional proposition
Direction: Transform the given conditional proposition.
Conditional
Converse
Inverse
Contrapositive
Let p = A figure is a triangle.
q = It is a polygon.
If a figure is a triangle, then it is a polygon.
If a figure is a polygon, then it is a triangle.
If a figure is not a triangle, then it is not a
polygon.
If a figure is not a polygon, then it is not a
triangle.

Validity of an argument
An argument is formed when we try to connect
bits of evidence (premises) in a way that will force the
audience to draw a desired conclusion.
Examples:
1. The person who robbed the Mini-Mart drives a 1989 Toyota
Tercel. Gomez drives a 1989 Toyota Tercel. Therefore, Gomez
robbed the Mini – Mart.
PREMISES:
A. The person who robbed the Mini-
Mart drives a 1989 Toyota Tercel
B. Gomez drives a 1989 Toyota
Tercel.
CONCLUSION:
Therefore, Gomez robbed the
Mini-Mart.

Examples:
2. The person who drank my coffee left this fingerprint on the cup.
Gomez is the only person in the world who has this fingerprint.
Therefore, Gomez is the person who drank my coffee.
PREMISES:
A. The person who drank my coffee
left this fingerprint on the cup.
B. Gomez is the only person in the
world who has this fingerprint.
CONCLUSION:
Therefore, Gomez is the
person who drank my coffee.

Syllogism is a formal argument that is formed
by two statements and a conclusion which must be
true if the two statements are true.
 An argument is said to be valid if it is logically
impossible to reject the conclusion but accept all of
the evidence.
 An argument is said to be invalid if it is logically
possible for the conclusion to be false even though
every proposition is assumed to be true.

RULES OF INFERENCE SYMBOLS
Contrapositive Reasoning
(Modus Tollens)
Direct Reasoning
(Modus Ponens)
Disjunctive Syllogism
Transitive Syllogism
Rule of Simplification
Rule of Addition
Rule of Conjunction
FALLACIES SYMBOLS
Fallacy of the
Converse
Fallacy of the Inverse
Affirming the
Disjunct
Fallacy of the
Consequent
Denying a Conjunct
Improper
Transposition

Direction: Determine whether the following arguments are valid.
1. If the apartment is damaged, then the deposit won't be refunded.
The apartment isn't damaged.
Therefore, the deposit will be refunded.
2. If I had a hammer, I would fix the chair
I did not fix the chair.
Therefore, I have a hammer.
3. All bulldogs are mean-looking dogs.
All mean-looking dogs are good watchdogs.
Therefore, all bulldogs are good watchdogs.

Direction: Determine whether the following arguments are valid.
4. 5. 6. 7.
8. 9. 10.

LOGIC in general mathematics123456789011

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