CMSC 250 Homework 2 Fall 2001

Due Wed Sept 12 at the beginning of your discussion section

1. Apply De Morgan's laws to the following English statements.
   1. Pete did not go to the store and Rose did not go to the store.

   2. It's not the case that both Pete and Rose went to the store.

   3. Pete did not go to the store or Sally did go to the store.

2. Express and completely simplify the negation of the following statements using De Morgan's laws.
   1. $a\leq b$.

   2. $a < b$ or $a\geq c$.

   3. $a = b$ and $a \neq c$.

3. Write the truth table for the following statement:
   $(\sim p\to q)\wedge (q\to \sim p).$ This statement is equivalent to the following: $\sim p$ $q$ where the blank is a single symbol.

4. Write statements that are the converse, the inverse, and the contrapositive of the following statements:
   1. If p, then q.

   2. If I have a car, then I can drive.

   3. If the square of an integer is even, then the integer is even.

5. Use a truth table to show that $(p\to q)\wedge (q\to r)\wedge (r\to p) = (p\wedge q\wedge r) \vee (\sim p \wedge \sim q \wedge \sim r).$
6. Write this statement's truth table: $\sim ((p\leftrightarrow q)\wedge ((\sim p \wedge q) \vee (p \wedge \sim q)))$.

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