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## Here's the Solution to this Question

a) $\left( {p \to \left( {q \to r} \right)} \right) \to \left( {\left( {p \wedge q} \right) \to r} \right) = \overline {\left( {p \to \left( {q \to r} \right)} \right)} \vee \left( {\left( {p \wedge q} \right) \to r} \right) = \overline {\left( {\overline p \vee \left( {q \to r} \right)} \right)} \vee \left( {\overline {\left( {p \wedge q} \right)} \vee r} \right) = \overline {\left( {\overline p \vee \left( {\overline q \vee r} \right)} \right)} \vee \left( {\overline {\left( {p \wedge q} \right)} \vee r} \right) = \overline {\left( {\overline p \vee \overline q \vee r} \right)} \vee \left( {\overline p \vee \overline q \vee r} \right) = p \wedge q \wedge \overline r \vee \overline p \vee \overline q \vee r = \left( {p \vee \overline p \vee \overline q \vee r} \right) \wedge \left( {q \vee \overline p \vee \overline q \vee r} \right) \wedge \left( {\overline r \vee \overline p \vee \overline q \vee r} \right) = \left( {T \vee \overline q \vee r} \right) \wedge \left( {T \vee \overline p \vee r} \right) \wedge \left( {T \vee \overline p \vee \overline q } \right) = T \wedge T \wedge T = T$

Q. E. D.

b) 1) $\left( {p \wedge q} \right) \wedge \left( {\left( {q \wedge \neg r} \right) \vee \left( {p \wedge r} \right)} \right) = \left( {p \wedge q} \right) \wedge \left( {\left( {q \vee p} \right) \wedge \left( {q \vee r} \right) \wedge \left( {\neg r \vee p} \right) \wedge \left( {\neg r \vee r} \right)} \right) = \left( {p \wedge q} \right) \wedge \left( {\left( {q \vee p} \right) \wedge \left( {q \vee r} \right) \wedge \left( {\neg r \vee p} \right) \wedge T} \right) = \left( {p \wedge q} \right) \wedge \left( {\left( {q \vee p} \right) \wedge \left( {q \vee r} \right) \wedge \left( {\neg r \vee p} \right)} \right) = \left( {p \wedge q} \right) \wedge \left( {q \vee \left( {p \wedge r} \right)} \right) \wedge \left( {\neg r \vee p} \right) = \left( {\left( {p \wedge q \wedge q} \right) \vee \left( {p \wedge q \wedge p \wedge r} \right)} \right) \wedge \left( {\neg r \vee p} \right) = \left( {\left( {p \wedge q} \right) \vee \left( {p \wedge q \wedge r} \right)} \right) \wedge \left( {\neg r \vee p} \right) = \left( {p \wedge q} \right) \wedge \left( {T \vee r} \right) \wedge \left( {\neg r \vee p} \right) = \left( {p \wedge q} \right) \wedge T \wedge \left( {\neg r \vee p} \right) = \left( {p \wedge q} \right) \wedge \left( {\neg r \vee p} \right) = p \wedge q \wedge \neg r \vee p \wedge q \wedge p = p \wedge q \wedge \neg r \vee p \wedge q = p \wedge q \wedge \left( {\neg r \vee T} \right) = p \wedge q \wedge T = p \wedge q$

2) $\neg \left( {p \to \neg q} \right) = \neg \left( {\neg p \vee \neg q} \right) = \neg \neg p \wedge \neg \neg q = p \wedge q$

So, $\left( {p \wedge q} \right) \wedge \left( {\left( {q \wedge \neg r} \right) \vee \left( {p \wedge r} \right)} \right) = p \wedge q$ and $\neg \left( {p \to \neg q} \right) = p \wedge q$

Then

$\left( {p \wedge q} \right) \wedge \left( {\left( {q \wedge \neg r} \right) \vee \left( {p \wedge r} \right)} \right) = \neg \left( {p \to \neg q} \right)$

Q. E. D.

c) $\left( {\left( {p \vee q} \right) \wedge \left( {p \to r} \right) \wedge \left( {q \to r} \right)} \right) \to r = \overline {\left( {\left( {p \vee q} \right) \wedge \left( {p \to r} \right) \wedge \left( {q \to r} \right)} \right)} \vee r = \overline {\left( {p \vee q} \right)} \vee \overline {\left( {p \to r} \right)} \vee \overline {\left( {q \to r} \right)} \vee r = \overline {\left( {p \vee q} \right)} \vee \overline {\left( {\overline p \vee r} \right)} \vee \overline {\left( {\overline q \vee r} \right)} \vee r = \left( {\overline p \wedge \overline q } \right) \vee \left( {p \wedge \overline r } \right) \vee \left( {q \wedge \overline r } \right) \vee r = \left( {\overline p \wedge \overline q } \right) \vee \overline r \wedge \left( {p \vee q} \right) \vee r = \overline {\left( {p \vee q} \right)} \vee \overline r \wedge \left( {p \vee q} \right) \vee r = \left( {\overline {\left( {p \vee q} \right)} \vee \overline r } \right) \wedge \left( {\overline {\left( {p \vee q} \right)} \vee \left( {p \vee q} \right)} \right) \vee r = \left( {\overline {\left( {p \vee q} \right)} \vee \overline r } \right) \wedge T \vee r = \left( {\overline {\left( {p \vee q} \right)} \vee \overline r } \right) \vee r = \overline {\left( {p \vee q} \right)} \vee \overline r \vee r = \overline {\left( {p \vee q} \right)} \vee T = T$

Q. E. D.