elementary logic.pptx

“If it was so, it might be; and if it
were so, it would be; but as it
isn’t, it ain’t. That’s logic.”
- Lewis Carroll, Alice in
Elementary
Mathematical
Logic

Table of
contents
Statement
01
Negation
02
Conjunction
03
Disjunction
04

Table of
contents
Conditional
05
Biconditional
06
Logical
Equivalence
07

Directions: Identify if the
following sentences are
STATEMENT or NOT
STATEMENT and determine
its truth value (that is you
should be able to decide
whether the statement is
TRUE or FALSE). In your
Am I Statement or Not?

1. Happy Birthday!
2. Some books have hard covers.
3. No Philippine presidents were naturally born in Korea.
4. My dog loves me.
5. Some cats aren't mammals.
6. Juan is intelligent.
7. Go get the milk.
8. All dogs are Labrador Retrievers.
9. Can we be friends?
10. Today is Tuesday. 5

1. Happy Birthday! – not statement
2. Some books have hard covers. -statement
3. No Philippine presidents were residents of Korea. -
statement
4. My dog loves me. -not statement
5. Some cats aren't mammals- statement
6. Juan is intelligent-not statement
7. Go get the milk.-not statement
8. All dogs are Labrador Retrievers.- statement
9. Can we be friends?- not statement
10. Today is Tuesday.- statement
6

2. Some books have hard covers. –statement - TRUE
3. No Philippine presidents were naturally born in Korea.
–statement- TRUE
5. Some cats aren't mammals- statement - FALSE
8. All dogs are Labrador Retrievers.- statement-
FALSE
10. Today is Tuesday.- statement-FALSE
7

 What can you say about the
sentences?
 How did you know that a
sentence is a statement or not?
 How would you define a

I. What is
Logic?
-is the science of correct
reasoning.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

Statement
01
is any declarative
sentence that is either
true (T) or false (F), but
not both

We refer
to T or F as the
truth value of the
statement.
Statements are
usually denoted by
lower case letters
(for example: p, q,
r,s …)

The sentence “Stand up”
or “Follow the leader”
cannot be used as
statements.
These are commands
instead of a declarative
sentence and therefore
cannot be classified as
true or false

Examples of
statements:
 𝑝: 1 + 1 = 2 (TRUE)
 𝑞: 2 + 3 = 6 (FALSE)
 𝑟: Today is Thursday.
 s: January has 31
days.
Examples of NOT
statements:
 𝑥+𝑦
 Happy Birthday!
 Tweet me. (command)
 Can we be friends?
(question)
 Juan is intelligent.
(opinion)

Let p be the sentence
“Manila is the capital in the
Philippines.” Let q be the
sentence “The world is flat.”
These are both statements
because we can assign to them a
truth value. In this case, p is true
and q is false.

Simple Statement?
Compound Statement?

A simple
statement is a
statement that
conveys a
single idea.
Examples:
 Today is March 16,
2020. (Simple statement)
 Metro Manila is under
community
quarantine. (Simple
statement)

A compound
statement is a
statement that
conveys two or
more ideas.
Examples:
Today is March 16,
2020 and Metro Manila
is under community
quarantine. (Compound
statement)
Today is not March 16,
2020 or yesterday was a
Sunday. (Compound
statement)

Statemen
t
Connective Symbolic
form
Type of
statement
not 𝑝 not ~𝑝 negation
𝑝 and 𝑞 and 𝑝 ∧ 𝑞 conjunction
𝑝 or 𝑞 or 𝑝 ∨ 𝑞 disjunction
If 𝑝, then 𝑞 If … then 𝑝 → 𝑞 conditional
𝑝 if and
only if 𝑞
if and only if 𝑝 ↔ 𝑞 Biconditional
Logic Connectives and Symbols

Negation
02
The negation of a
statement p, denoted by
~p (read as “not p”) is
the statement whose
truth value
is the opposite of the
truth value of p.

Exampl
e
p= Manila is not the
capital city of the
Philippines
p= Manila is the
capital city of the
Philippines

✘ Activity : Negate me!
Write the negation of each statement:
DIRITSO NA ANSWER
1. I want a car and a motorcycle.
2. My cat stays outside or it makes a mess.
3. I've fallen and I can't get up.
4. You study or you don't get a good grade.
5. It is raining and it isn't snowing.

Example:
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
a) Today is not Friday and I am going to a movie.
b) I am going to the basketball game and I am not going to a movie.
c) I am going to the movie if and only if it is raining.
d) If today is Friday, then I am not going to a movie.

Consider the following simple statements.
𝒔: I am not going to the basketball game.
Write the following compound statements in symbolic form.
SOLUTION:
a) Today is not Friday and I am going to a movie.
∼ 𝑝 ∧ 𝑟 Answer: ∼ 𝑝 ∧ 𝑟
b) I am going to the basketball game and I am not going to a movie.
∼ 𝑠 ∧ ∼ 𝑟 Answer: ∼ 𝑠 ∧∼ 𝑟
c) I am going to the movie if and only if it is raining.
𝑟 ↔ 𝑞 Answer: 𝑟 ↔ 𝑞
d) If today is Friday, then I am not going to a movie.
𝑝 → ∼ 𝑟 Answer: 𝑝 → ∼ 𝑟

✘ Activity 9: Translate me!
Direction: Consider the following simple
statements. Translate the following symbolic form
in compound statements.
𝒔: I am not going to love my ex anymore.

𝒔: I am not going to love
my ex .
1. (~p ↔ ~q)
2. (s∧~q)
3. (r ↔ ~p)
4. (s → ~q)
5. (~q ∧ p)
6. (~p∨ q)
7. (p ↔ ~p)
8. (s ↔ ~r)
9. (~r ∨ s)
10. (~q ∨ s)

✘ Application: Construct me!
Directions: Construct your own 4 simple or compound
statements labeling it with p, q, r and s respectively.
Translate the symbolic form given below using the
statements you constructed.
✘ 1. (s∧~q)
✘ 2. ~(~q∧r)
✘ 3. ~(p∨q)
✘ 4. (p∧~q)
✘ 5. (~s∧q)

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
values of its simple statements and its connectives.

Conjunction, 𝑝 ∧ 𝑞
 𝑝 ∧ 𝑞 is true if BOTH 𝑝 and 𝑞 are true. Otherwise, it is false.

Disjunction, 𝑝 ∨ 𝑞
 𝑝 ∨ 𝑞 is true if at least one statements (𝑝, 𝑞 or both) is true. It is false only
if both statements are false.

Examples: Negating simple statements.
𝑝: 1 + 1 = 2, ~𝑝: 1 + 1 ≠ 2
𝑞:June 26, 2020 is a Friday. ~𝑞:June 26, 2020 is not a Friday.
𝑟:The sun is bigger than the moon. ~𝑟:The sun is not bigger than the moon.
𝑞:I am not a Libra. ~𝑞:I am a Libra.
Negation, ~𝒑
 If 𝑝 is true, ~𝑝 is false
 If 𝑝 is false, ~𝑝 is true

Disjunction Conjunction
Negation
p q p∨q
T T T
T F T
F T T
F F F
p q p ∧ q
T T T
T F F
F T F
F F F
p q ~p ~q
T T F F
T F F T
F T T T
F F T T

“This is a quote, words
full of wisdom that
someone important said
and can make the
reader get inspired.”
—Someone Famous

Mercury
Mercury is the closest planet
to the Sun and the smallest
one in the Solar System. It’s a
bit larger than the Moon
Maybe you need to
divide the content
Venus
Venus has a beautiful name
and is the second planet from
the Sun. It’s a hot place—even
hotter than Mercury
01 02

You could use
three columns,
why not?
Mars
Despite being red, Mars is
actually a cold place full of
iron oxide dust
Jupiter
Jupiter is a gas giant and
the biggest planet in the
Solar System
Venus
Venus has a beautiful
name and is the second
planet from the Sun

Awesome
words
Because key words are great for
catching your audience’s attention

To modify this graph, click on it, follow the link,
change the data and paste the new graph here
Use a graph to
show your data
Jupiter
Jupiter is a gas giant, and
Solar System
Venus
planet from the Sun

Use a graph to
show your data
To modify this graph, click on it, follow the link,
change the data and paste the new graph here
20%
Despite being red, it is
actually a cold place
80%
It’s the biggest planet in
the Solar System

Sometimes,
reviewing
concepts
is a good idea
Mars
Despite being
red, Mars is a
cold place
Jupiter
It’s the biggest
planet in the
Solar System
Saturn
Saturn is the
ringed one and a
gas giant
Neptune
Neptune is the
farthest planet
from the Sun
Mercury
Mercury is the
closest planet to
the Sun
Venus
Venus has a
beautiful name,
but it’s very hot

What about four
columns?
Mercury
Mercury is the closest planet
to the Sun
Venus
Venus has a beautiful name,
but it’s very hot
Jupiter
It’s the biggest planet in the
Solar System
Saturn
Saturn is the ringed one and a
gas giant

Or you can use
three columns
Venus
Venus has a beautiful name,
but it’s very hot
Saturn
Saturn is the ringed one and a
gas giant
Jupiter
It’s the biggest planet in the
Solar System

Infographics make
your idea
understandable
Mercury
Mercury is the closest
planet to the Sun
Venus
name, but it’s very hot
Jupiter
It’s the biggest planet
in the Solar System
Saturn
Saturn is the ringed
one and a gas giant

Mission,
philosophy &
purpose
You can enter a subtitle here
if you need it
02

Daily lesson
planning
Mars
Despite being red, Mars is
actually a cold place full of
iron oxide dust
Jupiter
Jupiter is a gas giant and
Solar System
Mercury
planet to the Sun and the
smallest of them all
Venus
planet from the Sun

Teaching is
the highest
form of
understanding
Mercury is the closest planet to
the Sun and the smallest one in
the Solar System—it’s only a bit
larger than the Moon

Other specific
aspects of our
education in a
simple table
Mass Diameter
Mercury 0.06 0.38
Mars 0.11 0.53
Saturn 95.2 9.4
Jupiter
Jupiter is a gas giant, and the biggest
planet in the Solar System

This is a map
Mercury
planet to the Sun
Venus
Jupiter
It’s the biggest planet
in the Solar System

Meet our teaching
members
Clara
Here you could talk a
bit about this person
Ellie
Will

A timeline always
works well
2015 2017 2020 2023
Phase 1
It’s the biggest
planet in the
Solar System
Phase 2
Mercury is the
smallest one in
the System
Phase 3
Venus is the
second planet
from the Sun
Phase 4
Despite being
red, it’s actually a
cold place

Or try this
timeline
2015
2017
2020
2022
Mars
Mars is actually a
cold place full of
iron oxide dust
Mercury
Mercury is the
smallest planet in
the System
Jupiter
Jupiter is the
biggest planet in
the Solar System
Venus
It has a beautiful
name and is the
second planet

Big numbers catch your audience’s attention
7,498,300

Here are some
percentages
59%
Saturn
Saturn is the ringed one
and a gas giant
Venus
80%

24h 37m
386,000 km
distance from
Earth to the Moon
is Jupiter’s rotation
period
333,000
Earths in total is
the Sun’s mass

Desktop
software
You can replace the image on
the screen with your own
work. Just delete this one, add
yours and center it properly

Tablet app
You can replace the image on the
screen with your own work. Just
delete this one, add yours and
center it properly

Mobile web
You can replace the image with your own work. Delete
this one, add yours and center it properly
Mars
Mars is a cold place full of
iron oxide dust
Mercury
It’s the smallest planet in
the Solar System
Jupiter
Jupiter is the biggest
planet of them all
Venus
Venus is the second
planet from the Sun

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