1. (3=7 5+3 = 8) (10+3 = 12 9+2= 10).

A. True

B. False

2. 1+1 = 3 if and only if 2+2 =5.

A. True

B. False

3. (5>6 1+1=3) 7>5

A. True

B. False

4. p T = T, where p is a proposition and T is tautology.

A. True

B. False

5. p (q r) (p q) r

A. True

B. False

6. (x) is equivalent to (x).

A. True

B. False

7. y(x,y) is equivalent to y(x,y)

A.True

B.False

8. y(x,y) is equivalent to xy(x,y)

A. True

B.False

9. p q p q

A. True

B. False

10.

A. True

B. False

11. Use the truth table to prove or disprove. (p q) (p q) p

12. Give a negation for y(x>y)" . Note that negation or NOT cannot be explicitly used in your answer.

Group of answer choices

A y(x <= y)

B xy(x<=y)

C y(x

D xy(x<=y)

13. Suppose (,y) means x+y=5, where x and y are real numbers. Determine the truth value for the following statement.

y(x,y)

A True

B False

14. (x,y) means 2xy=5, where x and y are real numbers. Determine the truth value for the following statement.

yx(x,y)

A True

B False

15. Prove that if mn is even, then m is even or n is even, where m and n are integers. Hint: consider using contaposition.

16. Let F(x,y) be the statement x likes y where the domain consists of all people in the world.

Use quantifiers to express Not everyone likes Jack and Jill. Select all choices that are correct.

A x(F(x, Jack) F (x, Jill))

B x(F(x, Jack) F (x, Jill))

C x(F(x, Jack) F (x, Jill))

D x(F(x, Jack) F (x, Jill))

17. Use logical equivalence (the laws discussed in lecture) to show that (pq)(pq) is a tautology. Where p and q are propositions. Hint AB is equivalent to AB

18 Use proof by contradiction to show that sqrt2 is not a rational number. Note that you may use the theorem stated in question 15.

19. Additional file response. If you have written your proofs on paper, you may use this question to attach an image or a file (PDF or doc file preferred) as a file response. Please label answers clearly which questions are the answers for.