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

This question is really about how to correctly deny a claim that uses the word "all." A common trap is to swing too far and deny the opposite extreme instead of just denying "every single one."

1. All the mangoes in the basket are not good: this just restates a universal claim about badness instead of properly negating the original universal claim about goodness. It is not guaranteed by the original statement being false.
2. No mango in the basket is good: this says every mango is bad, which is a much bigger claim than "not all are good." If only one mango out of many is bad, the original statement is false, but this option would also be false, since some mangoes are still good here.
3. Some of the mangoes are good and some are not good: this assumes there is at least one good mango and at least one bad mango. But if every single mango turned out to be bad, the original claim is still false, yet this option fails too because it needs some good mangoes to exist.
4. There exists at least one mango in the basket that is not good: this is the weakest and most general statement among the four, it only claims that not every mango is good. It holds true whether one mango is bad or all of them are bad, so it is guaranteed no matter what the actual mix of good and bad mangoes turns out to be.

The correct option is "there exists at least one mango in the basket that is not good," since it is the only statement that must be true in every scenario where the original all-are-good claim fails.

Let's summarize:

• Negating "all A are B" gives "at least one A is not B," never "no A is B."
• Always test a candidate negation against the extreme case (here, all mangoes bad) to check it still holds.

So the necessarily true statement is that at least one mango in the basket is not good.