Logic-Statements-and-Quantifiers in Mathematics in the Modern World

1. Prayer

2. Always remember our
classroom rules!

3. Are you present?
Checking of Attendance

4. Make sure you are on your
seats according to the
seat plan

5. REVIEW!!!
1. When can you say that a congruence is true or not?
2. In what specific example can you give to prove that
modular arithmetic is applicable in everyday life?

6. Am I Statement or Not?
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 activity notebook. (3
minutes) 6

7. 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. 7

8. 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
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9. 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
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10. ✘ What can you say about the
sentences?
✘ How did you know that a sentence is
a statement or not?
✘ How would you define a statement?
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

11. GEC 126/113 Mathematics in the Modern World
Logical Statements and
Quantifiers

12. OBJECTIVES
✘ At the end of the lesson, the students should be able to:
a. Define and determine the truth value of a statement;
simple statements and compound statements
b. Construct a truth table for a given expression
c. Translate compound statements in symbolic form
d. Reflect on the domains of everyday application of logic
in life.
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13. 13

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

15. A statement is a declarative sentence
that is either true or false, but not
both.
*** Declarative sentence tells us something or gives
information THAT YOU CAN VERIFY if it is true or
false.
II. Mathematical Statements/Statements
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

16. It may not be necessary to determine whether a
sentence is true to determine whether it is a statement.
For instance, consider the following sentence.
Catriona Gray is Miss Universe 2017.
 In symbols, they are assigned in lower case letter:
p, q, r, s,…
II. Mathematical Statements/Statements
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

17. Examples of statements:
 𝑝: 1 + 1 = 2 (TRUE)
 𝑞: 2 + 3 = 6 (FALSE)
 𝑟: Today is Thursday.
 January has 31 days.
Examples of NOT
statements:
 𝑥 + 𝑦
 Happy Birthday!
 Tweet me. (command)
 Can we be friends?
(question)
 Juan is intelligent.
(opinion)
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

18. Simple Statement?
Compound Statement?

19. A simple statement is a statement that conveys a single idea.
A compound statement is a statement that conveys two or more ideas.
Examples:
 Today is March 16, 2020. (Simple statement)
 Metro Manila is under community quarantine. (Simple statement)
 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)
III. Simple Statements and Compound Statements
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

20. Statement 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
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

21. DEFINITION
NEGATION- a statement with the opposite
meaning and the opposite truth value
Statement: Manila is the capital of the
Philippines.
Negation: Manila is not the capital of the
Philippines.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

22. NEGATION
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

23. ✘ Activity 1: Negate me!
Write the negation of each statement:
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.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

24. 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.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

25. 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.
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
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

26. ✘ Activity 2: Translate me!
Direction: Consider the following simple
statements. Translate the following symbolic
form in compound statements.
𝒑: Today is Friday.
𝒒: It is raining.
𝒓: I am going to a movie.
𝒔: I am not going to love my ex anymore.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

27. 𝒑: Today is Friday.
𝒒: It is raining.
𝒓: I am going to a
movie.
𝒔: I am not going to
love my ex.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic
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)

28. ✘ 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)
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

29. 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.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

30. Conjunction, 𝑝 ∧ 𝑞
 𝑝 ∧ 𝑞 is true if BOTH 𝑝 and 𝑞 are true. Otherwise, it is false.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

31. 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.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

32. 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
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

33. Disjunction Conjunction Negation
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic
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

34. Reflect!!!
1.What are the things that we need to inculcate in our
minds to have a better logic or correct reasoning in
every situation encountered?
2. How is the learning of correct reasoning applicable
and useful in life after the lesson?

35. EVALUATION
Direction: Make a truth table for the given expression.
1. (~p∧q)
2. (p∧~q)
3. (p∧~q) ∨ ~p
4. (p∨q) ∧ (~q∧p)
5. (~p∧q) ∨ (~p∨q)
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

36. ASSIGNMENT
Research at least 10 situations in the field of your
chosen profession that logic or correct reasoning has
been applied.
M.N. ORTIZ GEC 113/126 Mathematics in the Modern World Module 2: Logic

37. SEE YOU NEXT MEETING!
If you have question about our discussion today, you may
see or send me an email at maria.ortiz@dmc.edu.ph
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