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Statements, Logical Connectives and Truth tables.

Here are some questions from a study packet for my final. I would like solutions and explanations so that I may practice before my big test. Thanks!

1. Write a simple sentence and then write the negation. Imagine situations where each statement is true or false. How does the truth value (true or false) of the negation statement compare with the truth value of the original statement?

2. Write two simple sentences and then write the compound statement you get from connecting the original two statements using the word "and." Imagine situations where each simple statement is true or false. How does the truth value (true or false) of the compound statement compare with the truth value of the original statements?
.

Here are symbols you may need:      U ∩  [ ] COPY AND PASTE!

3. Let p, q, and r be the following statements:
p: It is raining.
q: The clouds are dark.
r: The temperature is dropping.

Translate the following statements into English
(a) p  r (b) ~p  (q  r) (c) q  r (d) (  r   q)   p

4. Write in symbolic form using p, q, r, , , , , where p, q, r represent the following statements:

p: A dog is friendly.
q: A dog has a long tail.
r: A dog licks faces.

(a) If a dog has a long tail, then it is not friendly.
(b) If a dog is not friendly, then it does not lick faces.
(c) If a dog has a long tail, then either the dog is friendly or the dog licks faces.
(d) If a dog licks faces, then the dog is friendly and the dog has a long tail.

5. Fill the headings of the following truth table using p, q, , , , and .

p q (b)
T T F
T F T
F T T
F F T
p q (a)
T T F
T F T
F T F
F F F
(a) (b)

6. For each of the following conditionals, identify the antecedent and the consequent. Form the converse, inverse, and contrapositive.

(a) If I go to work, then I will get paid.
(b) Your yard will not look good if you don't mow the grass.

7. Rewrite ~ (p  ~q) using DeMorgan's laws.

8. (a) Translate the argument into symbolic form and (b) determine if the argument is valid or invalid. You may compare the argument to a standard form or use a truth table.

If it is raining, then we will close the window.
We closed the window.
----------------------------------------------------------------------
 It is raining.

9. Use an Euler diagram to determine whether the syllogism is valid or invalid

All doctors are helpful.
Some doctors are female.
--------------------------------
 Some helpful people are female.

10. Construct a truth table for ~q  (q  p).

Here are symbols you may need:      U ∩  [ ] COPY AND PASTE!

11. Write the negation for the statement below.

Someone in the family makes bread.

12. Let p, q, and r be the following statements:

p: Mary is on the bus.
q: April is in the car.
r: Stan is at the zoo.

Translate the following statement into English: (p   r)  q

13. Write the following compound statement in symbolic form

Let p: Today is Friday.
q: Tomorrow is not the day to go shopping.
If tomorrow is not the day to go shopping, then today is not Friday.

14. Construct a truth table for  (p  q)

15. Write the converse, inverse, and contrapositive of the following conditional statement

If the sun is shining, then it will not rain.

16. Determine whether the argument is valid or invalid.

A tree has green leaves and the tree produces oxygen.
This tree has green leaves
------------------------------------------------------------------------
 This tree produces oxygen.

17. Use Euler Diagrams to determine whether the following syllogism is valid or invalid.

All golfers have golf carts.
All members of the A club have golf carts.
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 All members of the A club are golfers

18. Determine the truth value of the statement q  [ r  (p  q)] when p is false, q is true, and r is true.

19. Determine the truth value of the following statement:

Rembrandt was a famous painter and all prime numbers are odd.

20. Use De Morgan's Laws to determine whether the two statements are equivalent

 (p  q),  p  q

21. Determine which, if any, of the three statements are equivalent.

a) If today is Monday, then tomorrow is Tuesday.
b) If today is not Monday, then tomorrow is not Tuesday.
c) If tomorrow is not Tuesday, then today is not Monday.

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