MATH 1010/0010

Linear Modeling and Logic Unit

4.7 Proposition and Truth Values

A proposition is a statement that is either true or false. A proposition must be in a complete sentence and must make a distinct assertion or denial.

• We are not always saying something that is true or false.
  • Please pass the salt.
  • Where is the closest Starbucks?
  • Stop at the red light.
• Sometimes we can observe if something is true or false.

  It is snowing outside.

  • If you look outside and it is snowing, the proposition is TRUE.
  • If you look outside and it is NOT snowing, the proposition is FALSE.
• We do not have to observe if a proposition is true or false. We can infer it from the circumstances.

Is each of the following a proposition? If so, is it true or false?

1. Entering “do a barrel roll” on Google makes the page rotate.

   Proposition. True

2. “We don't need no education.”

   Proposition. True

3. Are you hungry?

   Not a proposition.

4. No U.S. state allows its residents to vote from space.

   Proposition. False

5. "Open the pod bay doors, please, HAL."

   - DAVE BOWMAN (Keir Dullea) in 2001: A Space Odyssey (1968)

   Not a proposition.

Two propositions joined together by the word "and" is called a conjunction.

Conjunctions are only true if both propositions are true.

Two propositions joined together by the word "or" is called a disjunction.

Disjunctions are only false when both propositions are false.

Determine if the following statements are true or false:
6. Tennessee is in the south and Nashville is the capital.

   True

7. Austin is the capital of Texas or Knoxville is the capital of Tennessee.

   True

8. There are 12 inches in a yard and 1000 pounds in a ton.

   False

9. Verizon provides cell phone service or Comcast provides cable television service.

   True

10. Pork comes from pigs and veal comes from camels.

    False

11. Pellissippi State is a university or UTK is a community college.

    False





Conditionals


An If…Then statement is called a conditional statement because it proposes something to be true (the then part of the statement) on the condition that something else is true (the if part of the statement).

• The If part of the statement is called the hypothesis and the THEN part of the statement is called the conclusion.
• A conditional statement is defined to be true unless a true hypothesis leads to a false conclusion.



12. If we turn off the water in the shower, then the water will stop pouring.

    “We turned off the water in the shower" is the hypothesis and "the water will stop pouring" is the conclusion.

    
    Assuming we do turn off the water and it did stop, in which cases below I tell the truth when I made the conditional statement?
    • We turned off the water and the water stopped. (T→T)

      True

    • We turned off the water and the water did not stop. (T→F)

      False

    • We did NOT turn off the water and the water did not stop. (F→F)

      True

    • We did NOT turn off the water and the water did stop. (F→T)

      True

13. Determine if the following conditional statements are true or false.
    1. If February has 29 days, then it is a leap year.

    True

    2. If it is raining, then there are no clouds in the sky.

    False

    3. If the sun is made of rock, then 4 is a prime number.

    True