Use the following to answer questions 15 （使用以下回答问题15）.pdf

Chapter 1 Use the following to answer questions 1-5: In the questions below determine whether the proposition is TRUE or FALSE 1. 1 + 1 3 if and only if 2 + 2 3. Ans: True 2. If it is raining, then it is raining. Ans: True 3. If 1 0, then 3 4. Ans: True 4. If 2 + 1 3, then 2 3 − 1. Ans: True 5. If 1 + 1 2 or 1 + 1 3, then 2 + 2 3 and 2 + 2 4. Ans: False 6. Write the truth table for the proposition ¬(r → ¬q) ∨ (p ∧ ¬r). q r ¬(r → ¬q) ∨ (p ∧ ¬r) T T T T T T F T T F T F T F F T F T T T F T F F F F T F F F F F 7. (a) Find a proposition with the given truth table. p q ? T T F T F F F T T F F F (b) Find a proposition using only p ,q,¬, and the connective ∨ that has this truth table. Ans: (a) ¬p ∧ q, (b) ¬(p ∨ ¬q). Page 1 8. Find a proposition with three variables p , q, and r that is true when p and r are true and q is false, and false otherwise Ans: (a) p ∧ ¬q ∧ r. 9. Find a proposition with three variables p , q, and r that is true when exactly one of the three variables is true, and false otherwise Ans: (p ∧ ¬q ∧ ¬r) ∨ (¬p ∧ q ∧ ¬r) ∨ (¬p ∧ ¬q ∧ r). 10. Find a proposition with three variables p , q, and r that is never true Ans: (p ∧ ¬p ) ∨ (q ∧ ¬q) ∨ (r ∧ ¬r). 11. Find a proposition using only p ,q,¬ and the connective ∨ with the given truth table. p q ? T T F T F T F T T F F F Ans: ¬(¬p ∨ q) ∨ ¬(p ∨ ¬q). 12. Determine whether p → (q → r) and p → (q ∧ r) are equivalen