CMSC 250 Homework 1 ANSWERS Spring 2006

Due Wed Feb 1 at the beginning of your discussion section.

1. Convert the following sentences to logical expressions.
   1. $j \vee k \vee l$

   2. $(j \wedge k) \wedge \sim l$

   3. $j \vee k$

2. Determine which of the following are statements and which are not.
   1. Is a Statement.
   2. Is a Statement.
   3. Is a Statement.
   4. Is NOT a Statement.
   5. Is a Statement.

3. If the first column is known to be true, what else would need to be true in order to ensure that the second column is true?
   1. q must also be true
   2. nothing further is needed
   3. q must be false
   4. q must be false

4. Let $a$ and $b$ be statements. Construct the complete truth table for the following statement: $(a \vee b) \wedge (\sim a \vee \sim (a \vee b))$

   $a$	$b$	$c$	$a \vee b$	$\sim a$	$\sim (a \vee b$	$\sim a \vee \sim (a \vee b)$	$(a \vee b) \wedge (\sim a \vee \sim (a \vee b))$
   1	1	1	1	0	0	0	0
   1	1	0	1	0	0	0	0
   1	0	1	1	0	0	0	0
   1	0	0	1	0	0	0	0
   0	1	1	1	1	0	1	1
   0	1	0	1	1	0	1	1
   0	0	1	0	1	1	1	0
   0	0	0	0	1	1	1	0

5. Let $p$, $q$, and $r$ be statements. Construct the complete truth table for the following statement: $(\sim p \wedge q) \vee \sim r) \wedge p$

   $p$	$q$	$r$	$\sim p$	$\sim r$	$\sim p \wedge q$	$(\sim p \wedge q) \vee \sim r$	$((\sim p \wedge q) \vee \sim r) \wedge p$
   1	1	1	0	0	0	0	0
   1	1	0	0	1	0	1	1
   1	0	1	0	0	0	0	0
   1	0	0	0	1	0	1	1
   0	1	1	1	0	1	1	0
   0	1	0	1	1	1	1	0
   0	0	1	1	0	0	0	0
   0	0	0	1	1	0	1	0

6. Let $x$ and $y$ be statements. Construct the complete truth table for the following statement: $x \wedge ((y \vee \sim x) \vee (\sim x \wedge \sim y))$

   $x$	$y$	$\sim x$	$\sim y$	$y \vee \sim x$	$\sim x \wedge \sim y$	$(y \vee \sim x) \vee (\sim x \wedge \sim y)$	$((y \vee \sim x) \vee (\sim x \wedge \sim y)) \wedge x$
   1	1	0	0	1	0	1	1
   1	0	0	1	0	0	0	0
   0	1	1	0	1	0	1	0
   0	0	1	1	1	1	1	0

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