1. (4 Marks) Write the negation of each of the following statements.

(i) ∃x ∀y (P(x, y) → ¬Q(x, y)).
(ii) The tomato is green or it is not ready to eat.
(iii) If you practice solving mathematical problems, then you will have a
better understanding of the concepts.
(iv) ∀x ∀y [(x > y) ∧ (x − y > 0)].

2. (3 Marks) Construct the truth table for each of the following statements.

(a) p ∨ (¬q ∧ r)
(b) (¬p → q) → (q → ¬r)
(c) (p ←→ ¬q) → (p → q)

3. (2 Marks) Use two different methods to establish that the two propositions
A and B listed below are logically equivalent.

A : P → (Q ∨ R), B : (P ∧ ¬Q) → R

4. (3 Marks)

(a) Write the converse for the statement:
If x not equals y, then x4 not equals y4

(b) Write the contrapositive for the proposition:
If the electricity is out, then I cannot use the computer.

(c) Write the inverse of:
Snow is a sufficient condition for us not going to school.

5. (4 Marks) Which of the following are true? Show all your work.

(a) The proposition (p ←→ q) is logically equivalent to (p ∧ q) ∨ (¬p ∧ ¬q)
(b) The proposition q ←→ (p → ¬q) logically implies ¬q.
(c) The proposition (p → q) ∧ (q → r) ∧ (¬(p → r)) is a contradiction.
(d) The proposition (p → (q → r)) → ((p → q) → (p → r)) is a tautology. EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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