THE SEED MONTESSORI SCHOOL

GENERAL MATHEMATICS 11 Truth Values and Truth Tables

July 19, 2016

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Daily Routine

Objectives

Starter

Lesson Proper

Practice Exercises

Exit Card

At the end of the period, you are expected to be able to:

determine truth values of propositions; and

construct truth tables

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Tell whether the statement is true or false.

1. 1 is an even number.

2. If the sum of two angles is 180°, then they are supplementary.

3. All dogs are animals and all animals are dogs.

4. If you multiply two numbers, then the product is either positive or negative.

5. If you see lightning and you hear thunder, then it is raining somewhere near, and if it is raining somewhere near, then you see lightning and you hear thunder.

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Tell whether the statement is true or false.

1. 1 is an even number.

2. If the sum of two angles is 180°, then they are supplementary.

3. All dogs are animals and all animals are dogs.

4. If you multiply two numbers, then the product is either positive or negative.

5. If you see lightning and you hear thunder, then it is raining somewhere near, and if it is raining somewhere near, then you see lightning and you hear thunder.

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TRUE

TRUE

FALSE

FALSE

FALSE

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A teacher promised her student, “If you pass the final examination and you submit all the requirements, then you may join the graduation.” The student only passes the final examination; however, the teacher allows her to join the graduation.

Did the teacher break her promise?

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The truth value of a proposition is either TRUE (T) or FALSE (F), but not both.

In computer science and programming, 1 represents TRUE and 0 represents FALSE.

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A truth table is used to summarize all the possible combinations of truth values of a given proposition.

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1. Negation. The negation is true if and only if

the truth value of the proposition is false.

P: 1 is an even number. (F)

~P: 1 is not an even number. (T)

P ~P

T F

F T

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2. Disjunction. The disjunction is true if and

only if at least one of the disjuncts is true.

P Q P v Q

T T T

T F T

F T T

F F F

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P: I will clean the bathroom. Q: I will clean the kitchen. P v Q: I will clean either the bathroom or the

kitchen. Analysis: I lied only if I did not clean the

bathroom or I did not clean the kitchen. That is, F v F.

P Q P v Q

T T T

T F T

F T T

F F F

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3. Conjunction. The conjunction is true if and

only if both conjuncts are true.

P Q P ^ Q

T T T

T F F

F T F

F F F

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P: I will clean the bathroom.

Q: I will clean the kitchen.

P ^ Q: I will clean both the bathroom and the kitchen.

Analysis: I lied if I did not clean even just one of the two choices.

P Q P ^ Q

T T T

T F F

F T F

F F F

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4. Implication. The implication is true in all

cases EXCEPT when the consequent is false when the antecedent is true.

P Q P Q

T T T

T F F

F T T

F F T

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P: I will clean the bathroom.

Q: I will clean the kitchen.

PQ: If I will clean the bathroom, then I will clean the kitchen.

Analysis: I lied only if I did not clean the kitchen after I cleaned the bathroom.

P Q P Q

T T T

T F F

F T T

F F T

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1. Construct a truth table for: ~( P ^ Q ).

P Q

T

T F

F

T

T

F

F

F

F

F

T

T

T

T

F

P ^ Q ~(P^Q)

P Q

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2. Construct a truth table for: P v (Q~P)

T

T F

F

T

T

F

F

T

T

T

F

T

T

T

T

Q~P P v (Q~P) ~P

T

F

T

F

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A teacher promised her student, “If you pass the final examination and you submit all the requirements, then you may join the graduation.” The student only passes the final examination; however, the teacher allows her to join the graduation.

Did the teacher break her promise?

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First, symbolize the statement.

“If you pass the final examination and you submit all the requirements, then you may join the graduation.”

Let: P = The student passes the finals.

Q = The student submits all the requirements.

R = The student may join the graduation.

P ^ Q R ( )

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P Q R

Next, construct the truth table for (P^Q)R.

T

T F

F

T

T

F

T

T

T

F

F

T

T

T

( P ^ Q ) R P ^ Q

T

F

T

F

T

T

F T

T

F

F

F

F

F

T

F

F

F

F

F

F

T

T

T

T

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P Q R

But which one is his case?

T

T F

F

T

T

F

T

T

T

F

F

T

T

T

( P ^ Q ) R P ^ Q

T

F

T

F

T

T

F T

T

F

F

F

F

F

T

F

F

F

F

F

F

T

T

T

T

P = T; Q = F; R = T

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Write the conclusion.

Since the result is true, it means that the teacher did not break her promise.

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But what if the teacher did not allow him to join the graduation, did she break her promise?

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P Q R

Which one is this case?

T

T F

F

T

T

F

T

T

T

F

F

T

T

T

( P ^ Q ) R P ^ Q

T

F

T

F

T

T

F T

T

F

F

F

F

F

T

F

F

F

F

F

F

T

T

T

T

P = T; Q = F; R = F

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The result is still true. Meaning, the teacher’s decision not to allow him to join the graduation is still acceptable.

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Pair Work:

On your notebook, copy and answer numbers 1-10 of Firm Up, page 212.

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Answers:

1. F

2. F

3. F

4. F

5. F

6. T

7. T

8. T

9. F

10. T