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Logic

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Logic

Sentences, Statements and Truth Values

Unit 1

Geometry

Logic

- the science of reasoning

Mathematical sentences

declarative statements that may be either true or false (has a truth value).

The Ravena Library is on Main Street.

mathematical sentence

17 + 5 = 21

mathematical sentence

Go to your room.

Not a mathematical sentence

Do you have soccer practice today?

Not a mathematical sentence

14 – 8 = 6

mathematical sentence

Every parallelogram

Not a mathematical sentence (it’s a phrase)

Open Sentences

- sentences that contain a

variable

What is the variable of each sentence?

x + 2 = 8

the variable is x

He broke my pencil.

the variable is he

Jenny found it behind the sofa.

the variable is it

Domain

- the set of all elements that are possible replacements for the variable.

Solution set (truth set)

- the elements from the domain that make an open sentence true.

Open sentence:

It is the third month of the year.

It

Variable:

Domain:

{all the names of the months}

Solution (truth) set:

{March}

Closed Sentences

- a statement that can’t be judged as

true or false. There are no variables.

Negations

The negation of a statement always has the opposite truth value of the original statement.

Insert or take out the word ‘not’ to obtain the negation of a statement.

For example:

Statement: Neil Armstrong walked on the moon.

True

Negation: Neil Armstrong did not walk on the moon.

False

For example:

Statement: A duck is not a mammal.

True

Negation: A duck is a mammal.

False

Tell whether the following is a

mathematical sentence or not.

1. Thanksgiving is on the fourth Thursday in November.

Mathematical Sentence

2. Where did you go?

Not a Mathematical Sentence

3. Be quiet.

Not a Mathematical Sentence

4. y – 7 = 3y + 4

Mathematical Sentence

5. The first president of the United States was Abraham Lincoln.

Mathematical Sentence

What is the variable in the following open sentences.

5. She is tall.

6. 2y > 17

7. This country has the third largest population.

8. It is my favorite food.

Tell whether each sentence is true, false, or open.

9. The Statue of Liberty was given to the United States by France.

True

10. Tallahassee is a city in Montana.

False

11. 6x + 4 = 16

Open

Variable: x

12. 6(2) + 4 = 16

True

Find the truth set for each open sentence

using the replacement set

{Nevada, Illinois, Massachusetts, Alaska, New York}.

13. Its capital is Albany.

{New York}

14. It is on the east coast of the United States.

{Massachusetts, New York}

16. It is one of the last two states admitted to the United States.

{Alaska}

Write the negation of each sentence.

20. A stop sign is painted green.

F

T

A stop sign is not painted green.

21. There are 1,760 yards in a mile.

T

F

There are not 1,760 yards in a mile.

22. 14(2) – 16 = 12

T

F

14(2) – 16 ≠ 12

23. Today is not Thursday.

F

T

Today is Thursday.

Worksheet

Sentences, Statements, and Truth Values

1.01