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

2. For each of the following, use truth tables to determinewhether or not the two given statements are logically equivalentusing 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 whichshows 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 ”xknows y” where the domain of discourse for x and y is the set ofall people. Use quantifiers to express each of the followingstatements. (a) Alice knows everyone. (b) Someone knows George. (c)There is someone who nobody knows. (d) There is someone who knowsno one. (e) Everyone knows someone. 5. Again consider the predicateK(x, y) defined in Exercise 4. Negate in symbols the propositions(a), (c), and (e) from Exercise 4. Note: Of course, an easy way todo tis is to simply write ¬ in front of the answers forExercise

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

6. One more time, consider the predicate K(x, y) from Exercises4 and 5. Negate in smooth English the propositions (a), (c), and(e) from Exercise 4. Note: An easy way to do this is to simplywrite It is not the case that ... in front of each proposition.Don’t do that! Give the negation as a reasonably natural smoothEnglish sentence.

7. Three mathematicians are seated in a restaurant. The server:”Does everyone want coffee?” The first mathematician: ”I do notknow.” The second mathematician: ”I do not know.” The thirdmathematician: ”No, not everyone wants coffee.” The server comesback and gives coffee to the mathematicians who want it. Whichmathematicians received coffee? How did the waiter deduce whowanted coffee?