1. Construct a truth table for the following compound proposition: (p q) ? (p q)

2. Which of the following statements are tautologies?

(p ? q) ( p q)

(p ? q) ( (p ^ q)) (p q) ? (p ^ q)

(p ? q) ( q ? p)

[ q ^ (p ? q)] ? p

3. Show that (p ? r) ^ (q ? r) and (p V q) ? r are logically equivalent.

4.Show that (a^b^c^d^e) ? f and (a b c d e) ? f are not logically equivalent

5. Simplify the compound statement (p ^ (q r) ^ ((p ^ q) ? r)).

6. There are two tribes living on the island of Knights and Knaves: knights and knaves. Knights always tell truth and knaves always lie. You encounter two people A and B. What are A and B if A says If B is a knight then I am a knave!-.

7. Use inference rules to find out what relevant conclusion or conclusions can be drawn from this set of premises? Explain the rules of inference used to obtain each conclusion from the premises. I am going to hike this weekend.-

I will not go hiking on Sunday.-

If I go hiking on Sunday I will be tired on Monday.-

8. Write the following argument in symbolic form. Then establish the validity of the argument or give a counterexample to show that it is invalid.

If Dominic goes to the racetrack, then Helen will be mad. If Ralph plays cards all night, then Carmela will be mad. If either Helen or Carmela gets mad, then Veronica (their attorney) will be notified. Veronica has not heard from either of these two clients. Consequently, Dominic didn't make it to racetracks and Ralph didn't play cards all night.

9. Translate the following into symbolic form:

(i) Everybody likes him

(ii) Somebody cried out for help and called the police

(iii) Nobody can ignore her

10.Given the tree propositions p, q and r, construct truth tables for:

(i) (p ^ q) ? r

(ii) (p r) ^ q

(iii) p ^ ( q r)

(iv) p ? (¬q r)

(v) (p q) (r p).

11. prove that

1. Prove that (p ? q) ( p _ q)

2. Prove that (p ^ q) and (p ? q) are logically equivalent propositions.

3. Prove that (p q) (p ¬q)

12. Test the validity of the following arguments.

3. James is either a policeman or a footballer. If he is a policeman, then he has big feet. James has not got big feet so he is a footballer.

13. Solve the problems first and then consider some data on how children solved the problems found in the Children’s Solutions and Discussion of Problems and Exercises section.

1. Fill in the Venn diagram by listing the activities associated with each group.

2. Where would you place the following numbers in the Venn diagram:

15, 18, 20?

3. For this Venn Diagram, follow the directions and answer the questions that come after.

Lightly color the circle on the left yellow. Lightly color the circle on the right blue.

a. What does the yellow circle show?

b. How many children like soccer?

c. What does the blue circle show?

d. How many children like baseball?

e. What do you notice about Beth and Hector?

14. Using laws of set theory show that (A - B) - C = (A - C) - (B - C)

15. Let A, B, and C be sets. Show that

(A - C) n (C - B) = Ø

Draw Venn diagrams for the expression on the left side.

16. Select the Boolean expression that is not equivalent to x•x+x•x'

(a) x • (x + x') (b) (x + x') • x (c) x' (d) x

17. Select the expression which is equivalent to x • y + x • y • z

(a) x • y (b) x • z (c) y • z (d) x • y • z

18. Select the expression which is equivalent to (x + y) • (x + y')

(a) y (b) y' (c) x (d) x'

19. Select the expression that is not equivalent to x • (x + y) + y

(a) x • x' + y • (1 + x) (b) x' + x • y + y (c) x • y (d) y

20- Determine whether each of the following is a tautology, a contradiction or neither:

1. p ? (p q)

2. (p ? q) ^ ( p q)

3. (p q) (q p)

4. (p ^ q) ? p

5. (p ^ q) ^ ( p q)

Good luck

2. Which of the following statements are tautologies?

(p ? q) ( p q)

(p ? q) ( (p ^ q)) (p q) ? (p ^ q)

(p ? q) ( q ? p)

[ q ^ (p ? q)] ? p

3. Show that (p ? r) ^ (q ? r) and (p V q) ? r are logically equivalent.

4.Show that (a^b^c^d^e) ? f and (a b c d e) ? f are not logically equivalent

5. Simplify the compound statement (p ^ (q r) ^ ((p ^ q) ? r)).

6. There are two tribes living on the island of Knights and Knaves: knights and knaves. Knights always tell truth and knaves always lie. You encounter two people A and B. What are A and B if A says If B is a knight then I am a knave!-.

7. Use inference rules to find out what relevant conclusion or conclusions can be drawn from this set of premises? Explain the rules of inference used to obtain each conclusion from the premises. I am going to hike this weekend.-

I will not go hiking on Sunday.-

If I go hiking on Sunday I will be tired on Monday.-

8. Write the following argument in symbolic form. Then establish the validity of the argument or give a counterexample to show that it is invalid.

If Dominic goes to the racetrack, then Helen will be mad. If Ralph plays cards all night, then Carmela will be mad. If either Helen or Carmela gets mad, then Veronica (their attorney) will be notified. Veronica has not heard from either of these two clients. Consequently, Dominic didn't make it to racetracks and Ralph didn't play cards all night.

9. Translate the following into symbolic form:

(i) Everybody likes him

(ii) Somebody cried out for help and called the police

(iii) Nobody can ignore her

10.Given the tree propositions p, q and r, construct truth tables for:

(i) (p ^ q) ? r

(ii) (p r) ^ q

(iii) p ^ ( q r)

(iv) p ? (¬q r)

(v) (p q) (r p).

11. prove that

1. Prove that (p ? q) ( p _ q)

2. Prove that (p ^ q) and (p ? q) are logically equivalent propositions.

3. Prove that (p q) (p ¬q)

12. Test the validity of the following arguments.

3. James is either a policeman or a footballer. If he is a policeman, then he has big feet. James has not got big feet so he is a footballer.

13. Solve the problems first and then consider some data on how children solved the problems found in the Children’s Solutions and Discussion of Problems and Exercises section.

1. Fill in the Venn diagram by listing the activities associated with each group.

2. Where would you place the following numbers in the Venn diagram:

15, 18, 20?

3. For this Venn Diagram, follow the directions and answer the questions that come after.

Lightly color the circle on the left yellow. Lightly color the circle on the right blue.

a. What does the yellow circle show?

b. How many children like soccer?

c. What does the blue circle show?

d. How many children like baseball?

e. What do you notice about Beth and Hector?

14. Using laws of set theory show that (A - B) - C = (A - C) - (B - C)

15. Let A, B, and C be sets. Show that

(A - C) n (C - B) = Ø

Draw Venn diagrams for the expression on the left side.

16. Select the Boolean expression that is not equivalent to x•x+x•x'

(a) x • (x + x') (b) (x + x') • x (c) x' (d) x

17. Select the expression which is equivalent to x • y + x • y • z

(a) x • y (b) x • z (c) y • z (d) x • y • z

18. Select the expression which is equivalent to (x + y) • (x + y')

(a) y (b) y' (c) x (d) x'

19. Select the expression that is not equivalent to x • (x + y) + y

(a) x • x' + y • (1 + x) (b) x' + x • y + y (c) x • y (d) y

20- Determine whether each of the following is a tautology, a contradiction or neither:

1. p ? (p q)

2. (p ? q) ^ ( p q)

3. (p q) (q p)

4. (p ^ q) ? p

5. (p ^ q) ^ ( p q)

Good luck

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