Logic
Unit 1
Sentences, Statements and
Truth Values
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.
Variable: It
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
A stop sign is not painted green. T
21. There are 1,760 yards in a mile.
There are not 1,760 yards in a mile.
22. 14(2) – 16 = 12
14(2) – 16 ≠ 12
T
F
23. Today is not Thursday.
Today is Thursday.
F
T
T
F
Homework
Worksheet
Sentences, Statements,
and Truth Values
1.01