Negation of the Boolean statement (p ∨ q) ⇒ ((~ r) ∨ p) is equivalent to

Question

The number of different permutations of all the letters of the word "PERMUTATION" such that any two consecutive letters in the arrangement are neither both vowels nor both identical is:

Answer 1

To find the negation of the Boolean statement \((p \lor q) \Rightarrow ((\sim r) \lor p)\), we first need to understand the equivalence of implication in terms of logical operators.

The implication \(A \Rightarrow B\) can be rewritten as \(\sim A \lor B\). Therefore, the statement \((p \lor q) \Rightarrow ((\sim r) \lor p)\) can be expressed as:

\sim (p \lor q) \lor ((\sim r) \lor p)

To find the negation of the statement, we use the logical negation:

\sim (\sim (p \lor q) \lor ((\sim r) \lor p))

Using De Morgan's Laws and the double negation, this expression becomes:

Apply negation to the first component: p \lor q becomes \sim p \land \sim q.
The expression \sim ((\sim r) \lor p) becomes r \land \sim p (De Morgan's Laws).

Combine these results:

(\sim p \land \sim q) \land (r \land \sim p) = \sim p \land q \land r

Therefore, the negation of the original Boolean statement is equivalent to:

(\sim p) \land q \land r

This matches the option: (~p) ∧ q ∧ r

Answer 2