Question

Consider the following statements:
P: Suman is brilliant.
Q: Suman is rich.
R: Suman is honest.
The negation of the statement Suman is brilliant and dishonest if and only if Suman is rich can be expressed as which of the following:
(a)$-P\wedge \left( Q\leftrightarrow -R \right)$
(b)$-\left( Q\leftrightarrow \left( P\wedge -R \right) \right)$
(c)$-Q\leftrightarrow -P\wedge R$
(d)$-\left( P\wedge -R \right)\leftrightarrow Q$

Answer 1

Hint: Let Q be the statement that Suman is rich, P be the statement that Suman is brilliant, and R be the statement that Suman is honest. So, -R will be the statement that Suman is dishonest. Find the symbolic representation of Suman is brilliant and dishonest if and only if Suman is rich. Then find the negation of the resultant expression.

Complete step-by-step answer:
In this question, we are given the following:
Consider the following statements
P: Suman is brilliant.
Q: Suman is rich.
R: Suman is honest.
We need to find how the negation of the statement Suman is brilliant and dishonest if and only if Suman is rich can be expressed as.
Let Q be the statement that Suman is rich, P be the statement that Suman is brilliant, and R be the statement that Suman is honest.
So, -R will be the statement that Suman is dishonest.
So, the statement Suman is brilliant and dishonest if and only if Suman is rich can be expressed as the following:
$Q\leftrightarrow \left( P\wedge -R \right)$
So, the negation of the statement Suman is brilliant and dishonest if and only if Suman is rich can be expressed as the negation of the above expression in the following way:
$-\left( Q\leftrightarrow \left( P\wedge -R \right) \right)$
So, option (b) is correct.

Note: To solve this question, it is very important to know about the various logic symbols: how they are written and what they symbolize. Without this knowledge, you will not be able to solve the question.