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Which one of the following propositional forms is tautology?

  1. $p\ \wedge \sim (q\wedge p)$
  2. $p\ \wedge \sim q$
  3. $p\ \Rightarrow (p\wedge q)$
  4. $p\ \Rightarrow (p\vee q)$
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  1. $p\ \wedge \sim (q\wedge p)$ $\equiv$ $p\ \wedge (\sim q\ \vee \sim p) $ $\equiv$ $(p\ \wedge \sim q)\ \vee (p\ \wedge\sim p) $ $\equiv$ $(p\ \wedge \sim q)\ $
  2. $p\ \wedge \sim q$
  3. $p\ \Rightarrow (p\wedge q)$$\equiv$ $ \sim p\ \vee (p \wedge \ q) $$\equiv$ $ (\sim p\ \vee p)\ \wedge (\sim p \wedge \ q) $ $\equiv$ $(\sim p\ \wedge  q)\ $
  4. $p\ \Rightarrow (p\vee q)$$\equiv$ $ \sim p\ \vee (p \vee \ q) $$\equiv$  $ \sim p\ \vee p \vee \ q $$\equiv$ $ 1\vee \ q $$\equiv$  $1$

$\therefore$ Option D. is correct.

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