Question

'All the mangoes in the basket are good.'
If the above statement is false, then which one of the following statements is necessarily true?

• A. All the mangoes in the basket are not good.
• B. No mango in the basket is good.
• C. In the basket, some of the mangoes are good and some are not good.
• D. There exists at least one mango in the basket that is not good.

Hint:

The negation of "All A are B" is "At least one A is not B," not "No A is B."

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a universal statement, "All the mangoes in the basket are good," and told that this statement is false. We need the one statement that must be true as a result, no matter which particular mangoes actually turn out to be good or not good.

Step 2: Key Formula or Approach:
In logic, a universal statement of the form "All A are B" is negated by "There exists at least one A that is not B." This is a standard rule: the opposite of "every single one satisfies a property" is "at least one fails to satisfy it," it is not the much stronger claim that "every single one fails it."

Step 3: Detailed Explanation:
Applying this rule here: "All the mangoes in the basket are good" being false means "At least one mango in the basket is not good." This is exactly option (D).
Option (A), "All the mangoes in the basket are not good," rewords the subject but keeps the same universal (every single one) structure, so it is a different universal claim, not the true negation.
Option (B), "No mango in the basket is good," says every mango is bad. This is far stronger than what the negation guarantees. The original statement could be false even if 9 out of 10 mangoes are good and just 1 is not, and in that case option (B) would itself be false.
Option (C), "In the basket, some of the mangoes are good and some are not good," also over-specifies the situation. If it turns out that every mango in the basket is bad (zero good ones), the original "all are good" is still false, but option (C) would be false too, since it demands that some mangoes are good. So (C) is not guaranteed to hold in every case.
Only option (D) holds in every possible case where the original statement is false, whether just one mango is bad or every mango is bad.

Step 4: Final Answer:
The statement that is necessarily true is: "There exists at least one mango in the basket that is not good."
\[ \boxed{\text{Option (D)}} \]