Which one of the following Boolean expressions is a tautology ?

Question:

Which one of the following Boolean expressions is a tautology ?

  1. $(\mathrm{P} \vee \mathrm{q}) \wedge(\sim \mathrm{p} \vee \sim \mathrm{q})$

  2. $(P \wedge q) \vee(p \wedge \sim q)$

  3. $(P \vee q) \wedge(p \vee \sim q)$

  4. $(P \vee q) \vee(p \vee \sim q)$


Correct Option: , 4

Solution:

(1) $(p \vee q) \wedge(\sim p \vee \sim q) \equiv(p \vee q) \wedge \sim(p \wedge q) \rightarrow$

Not tautology (Take both $p$ and $q$ as $T$ )

(2) $(p \wedge q) \vee(p \wedge \sim q) \equiv p \wedge(q \vee \sim q) \equiv p \wedge t \equiv p$

(3) $(p \vee q) \wedge(p \vee \sim q) \equiv p \vee(q \wedge \sim q) \equiv p \vee c \equiv p$

(4) $(p \vee q) \vee(p \vee \sim q) \equiv p \vee(q \vee \sim q) \equiv p \vee t \equiv t$

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