Math 250 Fall 2001 Homework 1 solutions

1. 1. $p\wedge q$

   2. $(q\wedge p)\vee (q\wedge r)$

2. 1. Statement

   2. Not a statement

   3. Statement

   4. Statement

3. The student examples may be different.
   1. $((n < 6) \wedge (n \neq 4))\wedge (n > 2)$.

      n is between 2 and 6, but not equal to 4. Example: 3.

   2. $((n < 6) \vee (n\neq 4)) \wedge \sim (n > 2)$

      $((n < 6) \vee (n \neq 4)) \wedge (n \leq 2)$

      n is less than or equal to 2. Example: 1.

   3. $(n < 6) \wedge \sim (n\neq 4)$

      $(n < 6) \wedge (n = 4)$.

      n is equal to 4. Example: 4.

4. p	$\sim p$	$p\wedge \sim p$
   T	F	F
   F	T	F

5. p	q	$\sim p$	$\sim q$	$p \wedge \sim q$	$\sim p \wedge q$	$(p\wedge \sim q) \vee (\sim p\wedge q)$
   T	T	F	F	F	F	F
   T	F	F	T	T	F	T
   F	T	T	F	F	T	T
   F	F	T	T	F	F	F

6. p	q	r	$(p\wedge q)$	$(q\vee r)$	$(p \wedge q)\vee r$	$p\wedge (q \vee r)$
   T	T	T	T	T	T	T
   T	T	F	T	T	T	T
   T	F	T	F	T	T	T
   T	F	F	F	F	F	F
   F	T	T	F	T	T	F
   F	T	F	F	T	F	F
   F	F	T	F	T	T	F
   F	F	F	F	F	F	F

   They are not equivalent.

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