Logic Statements and Quantifiers by Nardito L. Mediana Jr.

Logic
StatementS
AND
Quantifiers
NARDITO L. MEDIANA JR.
MATHEMATICS IN THE MODERN WORLD

O
U
T
L
I
N
E
What is Logic?
Logic Statements
Simple Statements and
Compound Statements
Compound Statements
and Grouping Symbols
Quantifiers and Negation

The utility or mathematics goes
beyond the mundane. Mathematics
enables the development of codes
and ciphers that are useful to
individuals and to society
Source: https://ched.gov.ph/wp-content/uploads/2017/10/KWF-
Mathematics-in-the-Modern-World.pdf
core idea
You're too
young to
understand the
issues!
I'm 30 years old,
and you didn't
even address my
argument

Support the use of
mathematics in various aspects
and endeavors in life.
Source: https://ched.gov.ph/wp-content/uploads/2017/10/KWF-
Mathematics-in-the-Modern-World.pdf
lEARNING OUTCOME
You're too
young to
understand the
issues!
I'm 30 years old,
and you didn't
even address my
argument

Logic is the study of the
principles of correct reasoning.
The term "logic" came from
the Greek word logos, which is
sometimes translated as
"sentence", "discourse",
"reason", "rule", and "ratio".
What is Logic?
Source: https://philosophy.hku.hk/think/logic/whatislogic.php

Logic Statements
Every language contains different types of
sentences, such as statements, questions,
and commands. For instance,
"Is the test today?" is a question
"Go get the newspaper." is a command
"This is a nice car." is a an opinion
"Seoul is the capital of Korea" is a statement
of fact.

A Statement is a
declarative sentence
that is either true or
false, but not both true
and false
What is a Statement?

It may not be necessary to determine
whether a sentence is true to determine
whether it is a statement. For instance
consider the sentence below:
"American Shaun White won an Olympic
gold medal in speed skating."
You may not know if the sentence is true, but
you know that the sentence is either true or
it is false, and that it is not both true and
false. Thus, you know that the sentence is a
statement.

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Determine whether each
sentence is a statement.
a) Sorsogon is a province in
the Philippines.
b) How are you?
c) 9 + 2 is a prime number.
d) x + 1 = 5

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a) Sorsogon is a province in
the Philippines.
Answer: Sorsogon is a province
in Bicol Region in the
Philippines, so this sentence is
true and it is a statement.

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b) How are you?
Answer: The sentence "How
are you?" is a question; it is not
a declarative sentence. Thus, it
is not a statement.

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c) 9 + 2 is a prime number.
Answer: You may not know
whether 9 + 2 is a prime number;
however you do know that it is a
whole number greater than 1, so it
either a prime number or it is not.
The sentence is either true or it is
false, and it is not both true and
false, so it is a statement.

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d) x + 1 = 5
Answer: x + 1 = 5 is a
statement. It is known as an
open statement. It is true for
x = 4, and it is false for any other
values of x. For any given value
of x, it is true or false but not
both.

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sentence is a statement.
a) Open the door.
b) 7055 is a large number.
c) In the year 2027, the
president of the Philippines
is a woman.
d) x > 3.

A Simple Statement is a
statement that conveys a
single idea. A compound
statement is a statement that
conveys two or more ideas.
Simple Statements &
Compound Statements

Connecting simple statements with words or
phrases such as and, or, if-then, and if and
only if creates a compound statement.
For instance, "I will attend the meeting or I
will go to school." is a compound
statement. It is composed of the two simple
statements, "I will attend the meeting."
and "I will go to school." The word "or" is a
connective for the two simple statements.
George Boole used symbols such as p, q, r,
and s to represent simple statements and
the symbol to represent
connectives.

Logic Connectives and Symbols
Image Source: Mathematical Excursions, Third Edition (2013) by Richard N. Aufmann, Joanne S. Lockwood, Richard D. Nation, and
Daniel K. Clegg, pp. 114

The truth value of a simple
statement is either true (T) or false (F).
The truth value of a compound
statement depends on the truth value
of its simple statements and its
connectives.
The truth table is a table that shows
the truth value of a compound
statement for all possible truth values
of its simple statements.
Truth Value and Truth Tables

The negation of the statement "Today is
Friday." is the statement "Today is not
Friday."
In symbolic form, the tilde symbol is used
to denote the negation of a statement. If a
statement p is true, it's negations p is false,
and if the statement p is false, its negation
p is true.
The negation of the negation of statement, in
symbols ( p) is the original statement, thus,
( p) can be replaced by p in any
statement.

E
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A
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P
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Write the negation of each
statement:
a) Bill Gates has a yacht.
Answer: Bill Gates does not have
a yacht.
b) The fire engine is not red.
Answer: The fire engine is red

E
X
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C
I
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S
Write the negation of each
statement.
a) Avatar was selected as best
picture at the 82nd Academy
Awards ceremony.
b) The Queen Mary 2 is the
world's largest cruise ship.

Today is Friday and it is raining.
Consider the following simple statements:
p: Today is Friday.
q: It is raining.
r: I am going to a movie.
s: I am not going to the basketball game.
Write the following compound statements in
symbolic form:
1.
2. I am going to a basketball game or I am
going to a movie.
3. If it is raining, then I am not going to a movie.
Writing Compound Statements
in Symbollic Form

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1) Today is Friday and it is raining.
Answer: (p q)
2. I am going to a basketball game
or I am going to a movie.
Answer: (s r)
3. If it is raining, then I am not going
to a movie.
Answer: (s r)

(q p)
Consider the following simple statements:
p: Today is Friday.
q: It is raining.
r: I am going to a movie.
s: I am not going to the basketball game.
Write the following compound statements in
symbolic form:
1.
2. ( r s)
3. (s p)
Translating Symbolic
Statements to Verbal Statement

1. (q p)
Answer: It is raining and Today is
Friday.
2. ( r s)
Answer: I am not going to a movie
and I am not going to the basketball
game.
3. (s p)
Answer: I am not going to the
basketball game if and if Today is not
Friday.
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Compound Statements and Grouping Symbols
Image Source: Mathematical Excursions, Third Edition (2013) by Richard N. Aufmann, Joanne S. Lockwood, Richard D.
Nation, and Daniel K. Clegg, pp. 116
If a compound statement is written in symbolic form, then parentheses
are used to indicate which simple statements are grouped together. Table
below illustrates the use of parentheses to indicate groupings for some
statements in symbolic form.

Image Source: Mathematical Excursions, Third Edition (2013) by Richard N. Aufmann,
Joanne S. Lockwood, Richard D. Nation, and Daniel K. Clegg, pp. 116
If a compound statement is written as an English sentence, then a comma is used to
indicate which simple statements are grouped together. Statements on the same side of
a comma are grouped together.
If a compound statement is written as an English sentence, then the simple statements
that appear together in parentheses in symbolic form will all be on the same side of the
comma that appears in the English sentence.

Write (p q) r as an English sentence.
Let p, q, and r represent the following:
p: You get a promotion
q: You complete the training.
r: You will receive a bonus.
Perform the following:
1.
2. Write "If you do not complete the training,
then you will not get a promotion and you
will not receive a bonus." in symbolic form.
Translating Compound
Statements

1) Write (p q) r as an English
sentence.
Answer: Because p and the q statements both
appear in parentheses in the symbolic form,
they are placed to the left of the comma in the
English sentence.
Thus, the translation is: If you get a promotion
and complete the training, then you will
receive a bonus.
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2) Write "If you do not complete the
training, then you will not get a promotion
and you will not receive a bonus." in
symbolic form.
Answer: Because the not p and the not r
statements are both to the right of the comma in
the English sentence, they are grouped together
in parentheses in the symbolic form.
Thus, the translation is: q ( p r)
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Write (p q) r as an English sentence.
Let p, q, and r represent the following:
p: Kesha's singing style is similar to Ana's.
q: Kesha has messy hair.
r: Kesha is a rapper.
Perform the following:
1.
2. Write "If Kesha is not a rapper, then Kesha
does not have a messy hair and Kesha's
singing style is not similar to Ana's." in
symbolic form

The conjunction p q is true if and
only if both p and q are true, and it is
false if either p or q is false.
Truth Value of a Conjunction

The disjunction p q is true if and
only if both p is true, q is true, or both p
and q are true, and it is false if both p
and q are false.
Truth Value of a Disjunction

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statement is true or false
a) 5 is whole number and 5 is an
even number.
Answer: False, because 5 is not an
even number.
b) 2 is a prime number or 2 is an even
number
Answer: True, because 2 is prime
number and also an even number.

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statement is true or false:
a) 21 is a rational number and 21 is
a natural number.
b) 4 9
c) -7 -3

Existential Quantifiers - are used as prefixes
to assert existence of something. Examples of
existential quantifiers is the word some, and
the phrases "there exists" and "at least
one".
Universal Quantifiers - these include the
words none, no, all, and every. The universal
quantifiers "none" and "no" deny the
existence of something, where the universal
quantifiers "all" and "every" are used to assert
that every element of a given set satisfies
some condition.
Quantifiers and Negation

Quantified Statements and
their Negations

E
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A
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P
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Write the negation of the
following statements:
a) Some airports are open.
Answer: No airports are open.
b) All movies are worth the price.
Answer: Some movies are not worth
the price.
c) No odd numbers are divisible by 2.
Answer: Some odd numbers are
divisible by 2.

E
X
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Write the negation of the
following statements:
a) All bears are brown.
b) No smartphones are
expensive.
c) Some vegetables are not
green.

Thank
you!

References
Aufmann R.N.; Lockwood J.S.; Nation R.D.;
Clegg D.K., (2013). Mathematical Excursions,
Third Edition, pp. 111-119.
Dimasuay L.; Alcala J.; Palacio J. (2016).
General Mathematics for Senior High School.
C&E Publishing, Inc.
https://ched.gov.ph/wp-
content/uploads/2017/10/KWF-Mathematics-
in-the-Modern-World.pdf
https://philosophy.hku.hk/think/logic/whatislogi
c.php
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