Question

The dual of the statement $p \wedge \left[ {q \wedge \left( {p \vee q} \right) \wedge r} \right]$ is-
$p \vee \left[ {q \vee \left( {p \vee q} \right) \vee r} \right]$
$p \vee \left[ {q \vee \left( {p \wedge q} \right) \wedge r} \right]$
$p \vee \left[ {q \vee \left( {p \wedge q} \right) \vee r} \right]$
$p \vee \left[ {q \wedge \left( {p \wedge q} \right) \wedge r} \right]$

Answer 1

Hint:
Here, $ \vee $ is the sign of disjunction and $ \wedge $ is the sign of conjunction. To find the dual of the given statement, replace the conjunction sign $ \wedge $ with disjunction $ \vee $and disjunction sign $ \vee $ with conjunction sign $ \wedge $. And if there is any negation in the statement, do not change the negation of the statement.

Complete step by step solution:
The given statement is- $p \wedge \left[ {q \wedge \left( {p \vee q} \right) \wedge r} \right]$
Here, $ \vee $ is the sign of disjunction in which a proposition is true when either one or both of p and q are true and is false when both p and q are false.
But$ \wedge $ is the sign of conjunction in which a proposition is true when both p and q are true and is false when either or both of p and q are false.
We have to find the dual of this statement.
We know that the dual of as statement can be obtained by replacing the conjunction connective with disjunction connective and disjunction connective with conjunction connective. Then the two statements are called duals of each other and the connectives$ \vee $ and $ \wedge $ are also called duals of each other.
So here there are $3$ conjunction connectives in the given statement and $1$ disjunction connective.
So on replacing the connectives with each other, we get-
$ \Rightarrow $ $p \vee \left[ {q \vee \left( {p \wedge q} \right) \vee r} \right]$

The correct answer is option C.

Note:
In duality, each truth value is replaced by a false value and each false value is replaced by a truth value. So when truth values are changed, the connectives are also changed. We can also verify it by making the truth tables of both the statements. Then we will see that $p \wedge \left[ {q \wedge \left( {p \vee q} \right) \wedge r} \right] \to p \vee \left[ {q \vee \left( {p \wedge q} \right) \vee r} \right]$ is a tautology.