Consider the following statements P: Suman is brilliant Q: Suman is rich R : Suman is honest The neg

Q: Wait, why is 'dishonest' represented as $\sim R$? Isn't it just a different word?

In formal logic, we deal with binary truth values—true or false. If $R$ is the statement 'Suman is honest', then 'Suman is dishonest' is the exact opposite. If $R$ is true, 'dishonest' is false, and if $R$ is false, 'dishonest' is true. This is the definition of the negation operator $\sim$.

Q: Why can't I just distribute the negation inside the biconditional like $\sim(A \leftrightarrow B) \equiv \sim A \leftrightarrow \sim B$?

That is a common trap! The negation of a biconditional is not simply the negation of its parts. The biconditional $A \leftrightarrow B$ is true when $A$ and $B$ have the same truth value. Its negation is true when they have *different* truth values. You must keep the negation symbol outside the entire expression, as $\sim(A \leftrightarrow B)$.

Q: How did you know to swap the terms in the final step?

Great observation! The biconditional operator $\leftrightarrow$ is symmetric, meaning $A \leftrightarrow B$ is logically equivalent to $B \leftrightarrow A$. When looking at the options, if your derived expression doesn't match, always check if a simple rearrangement using this commutative property leads to one of the given choices.

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

Options:

(A) $\sim(Q \leftrightarrow (P \wedge \sim R))$

(B) $\sim Q \leftrightarrow \sim P \wedge R$

(C) $\sim(P \wedge \sim R) \leftrightarrow Q$

(D) $\sim P \wedge (Q \leftrightarrow \sim R)$

Sigma Percentile · Accepted Answer

### The Architecture of Thought: Deconstructing Logic

Welcome, future engineer. Today, we are not just solving a problem; we are peeling back the layers of human language to reveal the rigid, beautiful skeleton of mathematical logic.

Many students fear logic problems because they look like English sentences, but I want you to see them as equations waiting to be balanced. Let us embark on this journey to decode the statement: "Suman is brilliant and dishonest if and only if Suman is rich."

### Phase 1: The Atomic Foundation

Imagine you are a computer scientist building a circuit. You cannot process a complex sentence in one go; you must break it into "atomic" statements—the smallest units of truth.

We define our variables as follows: 
$P$: "Suman is brilliant"
$Q$: "Suman is rich"
$R$: "Suman is honest"

When the problem mentions "dishonest," it is not a new variable; it is the logical inverse of $R$. In the language of logic, we write this as $\sim R$. Now, our building blocks are ready.

### Phase 2: Building the Logical Structure

Next, we assemble these blocks. The phrase "brilliant and dishonest" connects $P$ and $\sim R$ with the conjunction "and."

In logic, "and" is the $\wedge$ operator. So, we have $(P \wedge \sim R)$.

Now, look at the phrase "if and only if." This is the biconditional operator, $\leftrightarrow$, which acts as a bridge between our first complex statement and the condition "Suman is rich" ($Q$).

Putting it all together, we get the symbolic representation:
$$(P \wedge \sim R) \leftrightarrow Q$$
This is the logical heart of the sentence.

### Phase 3: The Final Negation

Here is where the trap lies. The question asks for the **negation** of this entire statement.

A common mistake is to negate the individual parts, but that would change the meaning entirely. We must treat the entire biconditional as a single entity.

We wrap our expression in parentheses and place the negation operator $\sim$ outside:
$$\sim((P \wedge \sim R) \leftrightarrow Q)$$
This is the mathematically precise negation of the original claim.

### Phase 4: The Symmetry of Truth

Finally, we compare our result with the options. You might look at $\sim((P \wedge \sim R) \leftrightarrow Q)$ and feel a moment of panic because it does not look like the options.

But remember the beauty of the biconditional: it is a two-way street. The statement "$A$ if and only if $B$" is identical to "$B$ if and only if $A$."

Because of this symmetry, we can swap the positions of $Q$ and $(P \wedge \sim R)$ without changing the truth value of the biconditional. Thus, $\sim((P \wedge \sim R) \leftrightarrow Q)$ is logically equivalent to:
$$\sim(Q \leftrightarrow (P \wedge \sim R))$$

And there it is—the **final answer**. You have successfully navigated the logic, avoided the traps, and arrived at the truth. Keep this clarity with you; it is the hallmark of a brilliant mind.

Animated Solution for Mathematics - Sets and Relations: 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

Select Answer:

Visualized Solution

Identify Atomic Statements

Translate 'Dishonest'

Translate 'Brilliant and Dishonest'

Form the Complete Statement

Apply the Negation

Final Comparison with Options

The Sigma Insight: Representation of Sets

The Architecture of Thought

Phase 1

Phase 2

Phase 3

Phase 4

Similar Questions

Let A, B, C and D be four non-empty sets. The Contrapositive statement of “If $A \subseteq B$ and $B \subseteq D$ then $A \subseteq C$ ” is :

Let S be a non-empty subset of R. Consider the following statement : P : There is a rational number $x \in S$ such that $x > 0$ . Which of the following statements is the negation of the statement P ?

Let $A = {1, 2, 3, \dots, 10}$ and $B = {(m, n) : m, n \in A, m < n and g cd (m, n) = 1}$ . Then $n (B)$ is equal to :