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\textbf{Logic Exercises}
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Notation: In the following, to simplify the notation, I sometimes use $P \vee \sim Q$ to mean
$P \vee (\sim Q)$ and similarly with $\wedge$ and $\Rightarrow$; so $ \sim P \Rightarrow Q$ means $ (\sim P) \Rightarrow Q$.

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Exercise 1.  Determine if the following are theorems.

\qquad \qquad a. $ P \Rightarrow Q = (\sim Q \Rightarrow \sim P)$;

\qquad \qquad b. $ (\sim P) \Rightarrow Q = P \vee Q$;

\qquad \qquad c. $ \sim (P \vee \sim Q)  = (\sim P) \vee (\sim Q)$;

\qquad \qquad d. $ \sim (P \vee \sim Q)  = (\sim P) \wedge (\sim Q)$;

\qquad \qquad e. $ \sim (P \wedge \sim Q)  = (\sim P) \vee (\sim Q)$;

\qquad \qquad f. $ \sim (P \wedge \sim Q)  = (\sim P) \wedge (\sim Q)$;

\qquad \qquad g. $ (P \wedge Q) \vee R  = P \wedge (Q \vee R)$;

\qquad \qquad h. $ (P \vee Q) \wedge R  = P \vee (Q \wedge R)$.

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Definition: A tautology is a statement that has truth value $T$ no matter what are the truth value of the clauses making up the statement.

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Exercise 2. Determine which of the following are tautologies.

\qquad \qquad a. $ P \vee (\sim P)$;

\qquad \qquad b. $ P \wedge (\sim P)$;

\qquad \qquad c. $ \sim(P \vee (\sim P))$;

\qquad \qquad d. $ P \Rightarrow P $;

\qquad \qquad e. $ \sim(P \wedge (\sim P))$;

\qquad \qquad f. $ (P \Rightarrow Q) \vee Q$;

\qquad \qquad g. $ (P \Rightarrow Q) \vee (\sim Q)$;

\qquad \qquad h. $ (P \Rightarrow Q) \vee P$;

\qquad \qquad i. $ (P \Rightarrow Q) \vee (\sim P)$;

\qquad \qquad j. $ (P \Rightarrow Q) \wedge Q$;

\qquad \qquad k. $ (P \Rightarrow Q) \wedge (\sim Q)$;

\qquad \qquad l. $ (Q \vee (P \vee R)) \vee (\sim R)$;

\qquad \qquad m. $ (Q \wedge (P \wedge R)) \wedge (\sim R)$.







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