Understanding the Problem

We are given three statements:

• P: Suman is brilliant
• Q: Suman is rich
• R: Suman is honest

We need to find the negation of the statement: "Suman is brilliant and dishonest if and only if Suman is rich".

First, we express the given statement in logical form.

Step 1: Write the Given Statement in Logical Notation

"Suman is brilliant and dishonest" can be written as P ∧ (¬R), since dishonest is the negation of honest.

"if and only if Suman is rich" means this is equivalent to Q.

So, the entire statement is: (P ∧ (¬R)) ↔ Q

Using MathML: $(P\wedge \neg R)\leftrightarrow Q$

Step 2: Find the Negation of the Statement

We need the negation: ¬[(P ∧ (¬R)) ↔ Q]

Recall that for a biconditional statement A ↔ B, its negation is equivalent to A ⊕ B (exclusive OR), or more precisely, ¬(A ↔ B) ≡ (A ∧ ¬B) ∨ (¬A ∧ B)

Applying this here, with A = (P ∧ ¬R) and B = Q:

¬[(P ∧ ¬R) ↔ Q] ≡ [(P ∧ ¬R) ∧ ¬Q] ∨ [¬(P ∧ ¬R) ∧ Q]

Using MathML: $\left[\right(P\wedge \neg R)\wedge \neg Q]\vee [\neg (P\wedge \neg R)\wedge Q]$

Step 3: Simplify the Expression (Optional, for Understanding)

We can simplify the second part: ¬(P ∧ ¬R) ≡ (¬P ∨ R) by De Morgan's Law.

So the negation becomes: [(P ∧ ¬R) ∧ ¬Q] ∨ [(¬P ∨ R) ∧ Q]

This is a valid form of the negation, but we need to compare it to the given options.

Step 4: Compare with the Given Options

Let's look at the options (note: in the original text, Ù is used for ∧ (AND) and « is used for ↔ (if and only if). This is a font encoding issue. We will interpret them correctly:

• Option 1: ~ (P ∧ ~ R) ∧ Q (This is ¬(P ∧ ¬R) ∧ Q)
• Option 2: ~ (Q ↔ (P ∧ ~ R)) (This is ¬[Q ↔ (P ∧ ¬R)], which is the same as our negation ¬[(P ∧ ¬R) ↔ Q])
• Option 3: ~P ∧ (Q ↔ R) (This is ¬P ∧ (Q ↔ R))
• Option 4: ~ Q ↔ (~P ∧ R) (This is ¬Q ↔ (¬P ∧ R))

Option 2 directly matches the negation we found in Step 2.

Final Answer

The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" is ~ (Q ↔ (P ∧ ~ R)).

Using proper logical symbols: $\neg [Q\leftrightarrow (P\wedge \neg R\left)\right]$

Related Topics & Formulae

Key Logical Connectives

• Conjunction (AND, ∧): P ∧ Q is true only if both P and Q are true.
• Disjunction (OR, ∨): P ∨ Q is true if at least one of P or Q is true.
• Negation (NOT, ¬): ¬P is true when P is false, and false when P is true.
• Biconditional (IFF, ↔): P ↔ Q is true only when P and Q have the same truth value (both true or both false). Its negation is exclusive OR (XOR).

Important Equivalences and Laws

• De Morgan's Laws: $$\neg (P\wedge Q)\equiv \neg P\vee \neg Q$$ $$\neg (P\vee Q)\equiv \neg P\wedge \neg Q$$
• Negation of Biconditional: $$\neg (P\leftrightarrow Q)\equiv (P\wedge \neg Q)\vee (\neg P\wedge Q)$$

These laws are fundamental for simplifying and manipulating logical statements, especially when finding negations.