LOGICS - mathematics in the modern world

1. Lesson 3
ELEMENTARY LOGICS
ELEMENTARY LOGICS
MS. LAURENCE N. RUEDAS
MATHEMATICS IN THE MODERN WORLD

2. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Logic
• Logic is the study of correct thinking and
reasoning.
• It uses principles and methods to distinguish
valid arguments from those that are not.
• It is the foundation for expressing logical
methods used to prove theorems, design
computer software, and to solve mathematical
problems.
• Logic is a tool for working with complicated
statements.

3. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Propositions
• A statement or proposition is a declarative
sentence that is true or false but not both.
• Propositional variables such as p, q, r, s, t, etc.
are used to represent propositions.
Examples:
P: University of Northern Philippines is in
Vigan City.
Q: Light is faster than sound.
R: 1 + 3 = 4.
S: 7 is an even number.

4. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Simple and compound statement
• A simple statement is a statement that conveys
a single idea.
• A compound statement is a statement that
conveys two or more ideas.
• It is formed by connecting simple statements
with words and phrases such as and, or, if…
then, if and only if, etc.

5. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Simple or compound?
1. 20 is divisible by 4.
2. Taylor Swift is a singer and Stephen Curry
is a basketball player.
3. If a polygon has three sides, then it is a
triangle.
4. Mark goes to gym or stays at home every
Friday.

6. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Logical
Connectives

7. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Logical Connectives
Logical connective is a word or
symbol that joins two sentences to
produce a new one.
George Boole uses symbols such
as p, q, r, and s to represent simple
statements and the symbols ˄, ˅, ⁓,
, to represent connectives.
→ ↔

8. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Logical Connectives
Statements
Connectiv
e
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
→ implication/
conditional
p if and only
if q
if and only
if
p q
↔ biconditional

9. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Logical Connectives
Connectiv
e
Symbolic
Form
not ⁓ p
and p ˄ q
or p ˅ q
If…then p q
→
if and only
if
p q
↔
p: I review my lessons.
q: I play video games.
r: I go to the beach.
s: I get a reward.
a. q ˄ p
b. q ˅ r
c. ⁓ r
d. p s
→
e. s p
↔

10. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Logical Connectives
Connectiv
e
Symbolic
Form
not ⁓ p
and p ˄ q
or p ˅ q
If…then p q
→
if and only
if
p q
↔
a. John can program in C++
and he can program in
Java.
p: John can program in
C++.
q: John can program in
Java.
b. If x is an even number
then it is a multiple of 2.

11. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
The Truth Table

12. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
a. Negation
If a proposition p is true, then the
proposition ⁓p is false. However, if p is
false, then ⁓p is true
p: “I study at Philippine Normal
University.”
⁓p:

13. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
b. Conjunction
The conjunction of two statements p and q
denoted by p q is defined by the
⋀
following truth table.
The only condition for p q to be a true
⋀
p q p q
⋀
T T T
T F F
F T F
F F F

14. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
c. Disjunction
The disjunction of two statements p and q
denoted by p q is defined by the following
truth table.
This means that the disjunction of two
p q p q
⋁
T T T
T F T
F T T
F F F

15. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
d. Implication or Conditional
In a conditional statement, the truth of p
implies the truth of q. If p is true, then q
must be true. The only way that this can fail
(or be false) is when p is true while q is false.
The truth table of p q is given in the
→
following table.
p q p q
T T T
T F F
F T T
F F T

16. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
e. Biconditional
The biconditional statement p q, is defined
↔
by the following truth table.
p q p
T T T
T F F
F T F
F F T

17. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Summary of Truth Table
p q p q
⋀ p q
⋁ p q p
T T T T T T
T F F T F F
F T F T T F
F F F F T T

18. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Given the truth values of the propositions
A, B, C, and D. If A is true, B is false, C is
true, and D is false, give the truth value of
the following:
a. [(⁓A B) C ] D
⋀ → ⋁
p q p q
⋀ p q
⋁ p q p
T T T T T T
T F F T F F
F T F T T F
F F F F T T

19. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
Given the truth values of the propositions
A, B, C, and D. If A is true, B is false, C is
true, and D is false, give the truth value of
the following:
b. [ (A B) ⁓ C] [⁓ B ⁓ ( C
→ → ↔ → ⋀
D)]
p q p q
⋀ p q
⋁ p q p
T T T T T T
T F F T F F
F T F T T F
F F F F T T

20. • Vowels in the
alphabet
• Beautiful songs
• Prime numbers
less than 20
• Tall students
• Even numbers
between 1 and
A. Give the truth value if A is false, B is
true, C is false and D is true.
a. [(C B) ⁓ C] [B (C A)]
→ → ↔ → ⋀
b. [(D B) (A C)] B
⋀ ⋁ ⋀ ⋀
Assignment