Make a truth table to prove your answer is correct for each of the following:

2. Find a way to write P & Q without using &, but using ~ and instead, together with whatever brackets you need. Then find a way to write P Q without using, but using ~, & and brackets instead. Use tables to prove that your answer is correct.

3. For temporary purposes, use "*" as a symbol for exclusive "or." Construct a table to show the meaning of the connective. 4. Using the connective "*" from the previous exercise, use tables to investigate whether "~" distributes over "*"; that is, whether "~(P * Q)" says the same thing as "~P * ~Q."12

5. "*" is not one of our standard connectives. Suppose you wanted to express the content of "P * Q," but were restricted to using the connectives "~," "&" and "." How would you do it? How could you build up a complex sentence using "~", "&" and "" that says the same thing as "P * Q"? Explain with the help of a table.

6. Does "~" distribute over ""? Use a truth table to compare "~(P Q)" with "~P ~Q" to find out.

7. Use a truth-table to show that "P Q" means the same thing as "(~P Q) & (~Q P)."

8. The converse of "P Q" is "Q P." Use truth-tables to show that "P Q" does not mean the same thing its converse "Q P," but does mean the same thing as its contrapositive "~Q ~P."