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

• A. $~[q\leftrightarrow (p\wedge ~r\left)\right]$
• B. $~q\leftrightarrow p\wedge r$
• C. $~(p\wedge r)\leftrightarrow q$
• D. $~p\wedge (q\leftrightarrow r)$

Answer 1

The correct option is A

$~[q\leftrightarrow (p\wedge ~r\left)\right]$

Step 1. Given Statements :

p : Suman is brilliant

q : Suman is rich

r : Suman is honest;

t : “Suman is brilliant and dishonest if and only if Suman is rich”

Step 2. To find ~ t :

$$\begin{gathered} t:(p\wedge ~r)\leftrightarrow q \\ \therefore ~t:~\left[\right(p\wedge ~r)\leftrightarrow q]\dots \left(1\right) \\ \because ~(p\to q)=p\wedge ~qandp\leftrightarrow q=(p\to q)\wedge (q\to p) \\ \Rightarrow ~(p\leftrightarrow q)=~(p\to q)\vee ~(q\to p)=(p\wedge ~q)\vee (q\wedge ~p)\dots \left(2\right) \end{gathered}$$

Step 3. Conclusion from (1) and (2) :

$~t:\left[\right(p\wedge ~r)\wedge ~q]\vee [q\wedge ~(p\wedge ~r)$which is equivalent to $~[q\leftrightarrow (p\wedge ~r\left)\right].$

Hence, the correct option is A.