Question

The negation of the statement "The number is an odd number if and only if it is divisible by 3."

• A. The number is an odd number but not divisible by 3 or the number is divisible by 3 but not odd.
• B. The number is not an odd number iff it is not divisible by 3.
• C. The number is not an odd number but it is divisible by 3.
• D. The number is not an odd number or is not divisible by 3 but the number is divisible by 3 or odd.

Hint:

Logic Tip:In logic, the conjunctions but and and ($\wedge$) serve the exact same mathematical function.Always memorize the biconditional negation rule: $\sim(p \leftrightarrow q) \equiv (p \wedge \sim q) \vee (q \wedge \sim p)$.

Answer 1

The Correct Option is A

Concept:
Mathematical Logic - Negation of a Biconditional Statement.

Step 1: Identify the core logical propositions.
Let $p$: "The number is an odd number."
Let $q$: "The number is divisible by 3."

Step 2: Formulate the logical expression for the original statement.
The phrase "if and only if" corresponds to the biconditional operator ($\leftrightarrow$). Thus, the original statement is represented logically as $p \leftrightarrow q$.

Step 3: State the standard logical equivalence for the negation.
The negation of the statement is written as $\sim(p \leftrightarrow q)$. According to the rules of logical equivalence, this expansion is evaluated as $(p \wedge \sim q) \vee (q \wedge \sim p)$.

Step 4: Translate the first part of the logical equivalence.
The first expression $(p \wedge \sim q)$ translates to "$p$ AND NOT $q$". In English, this is written as "The number is an odd number AND it is NOT divisible by 3". (Note: the word 'but' is logically equivalent to 'and').

Step 5: Translate the second part, combine them, and finalize the statement.
The second expression $(q \wedge \sim p)$ translates to "$q$ AND NOT $p$", which simply means "The number is divisible by 3 AND it is NOT an odd number". These two distinct parts are connected by the main logical OR operator ($\vee$).
When we synthesize these translated phrases into a single, coherent sentence, we construct the final compound statement: "The number is an odd number but not divisible by 3 OR the number is divisible by 3 but not odd.
"Comparing this derived statement against the given choices reveals that it is an exact match for option A.