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 false version of "All A are B" only tells you "at least one A is not B," nothing stronger.

Answer 1

The Correct Option is D

The original claim is a universal one: "every mango in the basket is good." To find what must follow when a universal claim is false, it helps to test each option against a possible real basket of mangoes and see if it always holds.

1. All the mangoes in the basket are not good: Picture a basket with 9 good mangoes and 1 bad one. The original claim ("all are good") is false here, since one is bad. But this option says every mango is bad, which is not true for our basket of mostly good mangoes. So this option can fail even when the original statement is false.
2. No mango in the basket is good: Using the same 9-good-1-bad basket, this option is also false, since 9 mangoes are in fact good. So it, too, is not guaranteed.
3. In the basket, some of the mangoes are good and some are not good: This holds for the 9-good-1-bad basket, but now picture a basket where all 10 mangoes are bad. The original claim is still false here (not all are good), yet this option is false, because there are no good mangoes at all in that basket. So it cannot be guaranteed either.
4. There exists at least one mango in the basket that is not good: In the 9-good-1-bad basket, this holds (the 1 bad one). In the all-10-bad basket, this also holds (every one of them is a mango that is not good). In fact, in any basket where "all mangoes are good" is false, there must be at least one non-good mango by definition of the claim being false.

Testing across different possible baskets shows that only the last option holds in every case where the original statement fails. So this is the one statement that is necessarily true.

Let's sum up:

• The false version of "all A are B" only guarantees "at least one A is not B," never a stronger claim about all or none of the A's.
• Building sample baskets that make the original statement false, but try to break each option, quickly rules out the too-strong choices.

So the statement that must be true is "there exists at least one mango in the basket that is not good," option (D).