Enter T or F depending on whether the formula is ambiguous (T) or not (F). (You must enter T or F -- True (true) and False (false) will not work.)

 1. $p \wedge q \rightarrow s$
 2. $p \vee (q \vee r)$
 3. $p \wedge q \wedge r \wedge s \wedge w$
 4. $p \vee q \vee r \vee s \wedge w$
 5. $p \wedge q \vee s$

Enter T or F depending on whether the formula is a conjunctive normal form or not. (You must enter T or F -- True (true) and False (false) will not work.)

 1. $(p \vee q) \wedge (\neg p \vee r \vee q) \wedge (t \vee w)$
 2. p
 3. $\neg (p \wedge q)$
 4. $p \wedge q$
 5. $r \leftrightarrow q$

Enter T or F depending on whether the formula is a disjunctive normal form or not. (You must enter T or F -- True (true) and False (false) will not work.)

 1. p
 2. $(p \wedge q) \vee (\neg p \wedge r \wedge q) \vee (t \wedge w)$
 3. $\neg (p \wedge q)$
 4. $\neg p$
 5. $p \wedge q$

What formula is a disjunctive normal form of $\neg(b \vee (a \wedge c))$?

A. $(\neg b \wedge \neg a) \vee (\neg a \wedge \neg c)$
B. $(\neg b \wedge \neg a) \vee (\neg b \wedge \neg c)$
C. $(\neg b \wedge \neg a) \vee (\neg b \wedge c)$
D. $(\neg b \wedge \neg a) \vee (b \wedge \neg c)$
E. $(b \wedge a) \vee (\neg b \wedge \neg c)$
F. $(\neg b \wedge \neg c) \vee (b \wedge \neg c)$
G. $(\neg b \wedge a) \vee (\neg b \wedge \neg c)$

Provide the justifications for the following transformation in disjunctive normal form at each step, using the equivalences listed above. We start with a formula in conjunctive normal form.

$(a \vee b ) \wedge (\neg a \vee d)$ $\equiv$
$((a \vee b) \wedge \neg a) \vee ((a \vee b) \wedge d)$ by $\equiv$ $(a \wedge \neg a) \vee (b \wedge \neg a) \vee (a \wedge d) \vee (b \wedge d)$ by $\equiv$ $(b \wedge \neg a) \vee (a \wedge d) \vee (b \wedge d)$ by (Conjunctive normal form)