**Unit 3 Exercises**

**1**. Which of the following sentences are propositions?
What are the truth values of those that are propositions?

- Richmond is the capital of Virginia.
- 2 + 3 = 7.
- Open the door.
- 5 + 7 < 10.
- The moon is a satelite of the earth.
*x*+ 5 = 7.*x*+ 5 > 9 for every real number*x*.

**2.** What is the negation of each of the following propositions?

- Norfolk is the capital of Virginia.
- Food is not expensive in the United States.
- 3 + 5 = 7.
- The summer in Illinois is hot and sunny.

**3**. Let *p* and *q* be the propositions

*p*: Your car is out of gas.

*q*: You can't drive your car.

Write the following propositions using *p* and *q*
and logical connectives.

a) Your car is not out of gas.

b) You can't drive your car if it is out of gas.

c) Your car is not out of gas if you can drive it.

d) If you can't drive your car then it is out of gas.

**4**. Determine whether each of the following implications is true or false.

a) If 0.5 is an integer, then 1 + 0.5 = 3.

b) If cars can fly, then 1 + 1 = 3.

c) If 5 > 2 then pigs can fly.

d) If 3*5 = 15 then 1 + 2 = 3.

**5. **State the converse and contrapositive of each of the following implications.

- If it snows today, I will stay home.
- We play the game if it is sunny.
- If a positive integer is a prime then it has no divisors other than 1 and itself.

**6. **Construct a truth table for each of the following compound propositions.

a) *p*
*p*
* *
c) (*p*
*q*)
*q*
* *e) (*p*
*q*)
( *q*
*p*)