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: State the general rule for negating universal statements.
The negation of "for all x, P(x)" is "there exists an x such that not P(x)". This is the standard result from predicate logic and applies directly here, where P(x) means "mango x is good".

Step 2: Test option (A) and (B).
"All the mangoes in the basket are not good" and "No mango in the basket is good" both claim that every mango is bad. That is a far stronger claim than "not all mangoes are good"; the original statement being false does not tell us that every mango is bad, only that at least one is not good. So both of these options go beyond what the negation guarantees.

Step 3: Test option (C).
"In the basket, some of the mangoes are good and some are not good" assumes there is a mixture of good and bad mangoes. But it is entirely possible that all the mangoes are bad, in which case there are no good mangoes at all, so this option is not guaranteed either.

Step 4: Test option (D).
"There exists at least one mango in the basket that is not good" is exactly the logical negation of "all mangoes are good". As soon as one mango in the basket is not good, the original universal statement becomes false, and this is the weakest, most general condition that must hold.

Final Answer:
The statement that is necessarily true is option (D): there exists at least one mango in the basket that is not good.