Question

1. Consider the implication: If it is snowing, then I will go cross country skiing. (a) Write the converse of the implication. (b) Write the contrapositive of the implication. (c) Write the inverse of the implication.

2. For each of the following, use truth tables to determine whether or not the two given statements are logically equivalent using truth tables. Be sure to state your conclusions. (a) p → (q ∧ r) and (p → q) ∧ (p → r) (b) (¬p ∧ (p → q)) → ¬q and T

3. Give a two-column proof in the style of Section 2.6 which shows the following symbolic argument is valid. (¬p ∨ q) → r s ∨ ¬q ¬t p → t (¬p ∧ r) → ¬s ∴ ¬q 4. Let K(x, y) be the predicate ”x knows y” where the domain of discourse for x and y is the set of all people. Use quantifiers to express each of the following statements. (a) Alice knows everyone. (b) Someone knows George. (c) There is someone who nobody knows. (d) There is someone who knows no one. (e) Everyone knows someone. 5. Again consider the predicate K(x, y) defined in Exercise 4. Negate in symbols the propositions (a), (c), and (e) from Exercise 4. Note: Of course, an easy way to do tis is to simply write ¬ in front of the answers for Exercise

4. Don’t do that! Give the negation with no quantifiers coming after a negation symbol.

6. One more time, consider the predicate K(x, y) from Exercises 4 and 5. Negate in smooth English the propositions (a), (c), and (e) from Exercise 4. Note: An easy way to do this is to simply write It is not the case that ... in front of each proposition. Don’t do that! Give the negation as a reasonably natural smooth English sentence.

7. Three mathematicians are seated in a restaurant. The server: ”Does everyone want coffee?” The first mathematician: ”I do not know.” The second mathematician: ”I do not know.” The third mathematician: ”No, not everyone wants coffee.” The server comes back and gives coffee to the mathematicians who want it. Which mathematicians received coffee? How did the waiter deduce who wanted coffee?

Answer 1

7)

First and Second mathematicians received coffee.

If he knew that he does not want coffee, then he could tell the waitress 'no', because he then knows that NOT everyone wants coffee. The only possibility that he doesn't know it when he is asked is that he does want coffee but does not know if the other two want some too. The same logic applies for the second professor. Only the third then has perfect knowledge and can say 'yes' if he also wants coffee, and 'no' if not.