Question

$ \sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$ is logically equivalent to
1.$ \sim p$
2. $p$
3. $q$
4. $ \sim q$

Answer 1

Hint: In this question, we are given to find the value of $ \sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$ logical equivalent. First, we’ll solve the starting bracket using de morgan's law. After applying the law, you see that the next step is in the form of distributive law apply and in step, there will be a term that can be neglected so you will get the answer as $ \sim p$.

Formula Used:
De-Morgan’s law –
$ \sim \left( {p \vee q} \right) \equiv \left( { \sim p \wedge \sim q} \right)$
Distributive law –
$p \wedge \left( {q \vee r} \right) \equiv \left( {p \wedge q} \right) \vee \left( {p \wedge r} \right)$
Negotiation law –
$p \vee \sim p \equiv T$
$p \wedge \sim p \equiv F$
Here, $ \sim $ is the negotiation logic operation or NOT
$ \vee = $ Disjunction logic operator (OR)
And $ \wedge = $ conjunction logic operator (AND)
Where $p,q,r$ are prepositions

Complete step by step Solution:
Given that,
$ \sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$
Using De-Morgan’s law, not ($A$ or $B$) = (not$A$) and (not$B$)
$ \equiv \left( { \sim p \wedge \sim q} \right) \vee \left( { \sim p \wedge q} \right)$
Using distributive law, $p \wedge \left( {q \vee r} \right) \equiv \left( {p \wedge q} \right) \vee \left( {p \wedge r} \right)$
$ \equiv \sim p \wedge \left( { \sim q \vee q} \right)$
Using Negotiation law,
$ \equiv \sim p$

Hence, the correct option is 1.

Note: The key concept involved in solving this problem is a good knowledge of Logical equivalence. Students must know that if the statement forms two statements that are logically equivalent, they are said to be logically equivalent. It is simple to demonstrate logical equivalence or inequity. Two forms are equivalent if and only if their truth values are the same, so we create a table for each and compare the truth values (the last column). Also, to prove and solve any logical equivalence apply the laws. Basically, all laws with their names should be on your tip.