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?
'All the mangoes in the basket are good.'
If the above statement is false, then which one of the following statements is necessarily true?
If the above statement is false, then which one of the following statements is necessarily true?
Show Hint
The false version of "All A are B" only tells you "at least one A is not B," nothing stronger.
Updated On: Aug 7, 2026
- All the mangoes in the basket are not good.
- No mango in the basket is good.
- In the basket, some of the mangoes are good and some are not good.
- There exists at least one mango in the basket that is not good.
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
We are told that the statement "All the mangoes in the basket are good" is false, and we must find what this false universal statement forces to be true.
Step 2: Key Formula or Approach:
In logic, a universal statement of the form "All A are B" is false exactly when its negation is true.
The negation of "All A are B" is "Some A are not B," which means "at least one A is not B." It is not the stronger claim "No A is B."
Step 3: Detailed Explanation:
Here A is "mangoes in the basket" and B is "good."
Since "All mangoes are good" is false, the negation "at least one mango in the basket is not good" must be true.
This negation does not tell us how many mangoes are bad, only that at least one is.
Option (A), "All the mangoes are not good," actually claims every mango is bad, which is a much stronger and separate statement than the negation, so it is not necessarily true.
Option (B), "No mango is good," says the same overly strong thing as (A) in different words, so it also fails.
Option (C) assumes there are both good and bad mangoes in the basket. This could be true in one scenario, but it is not forced: it is equally possible that every mango is bad, which would make (C) false while the original statement is still false. So (C) is not necessarily true.
Option (D), "There exists at least one mango in the basket that is not good," is exactly the logical negation of the original statement, so it must be true whenever the original is false.
Step 4: Final Answer:
The statement that is necessarily true is option (D). \[ \boxed{\text{Option (D)}} \]
We are told that the statement "All the mangoes in the basket are good" is false, and we must find what this false universal statement forces to be true.
Step 2: Key Formula or Approach:
In logic, a universal statement of the form "All A are B" is false exactly when its negation is true.
The negation of "All A are B" is "Some A are not B," which means "at least one A is not B." It is not the stronger claim "No A is B."
Step 3: Detailed Explanation:
Here A is "mangoes in the basket" and B is "good."
Since "All mangoes are good" is false, the negation "at least one mango in the basket is not good" must be true.
This negation does not tell us how many mangoes are bad, only that at least one is.
Option (A), "All the mangoes are not good," actually claims every mango is bad, which is a much stronger and separate statement than the negation, so it is not necessarily true.
Option (B), "No mango is good," says the same overly strong thing as (A) in different words, so it also fails.
Option (C) assumes there are both good and bad mangoes in the basket. This could be true in one scenario, but it is not forced: it is equally possible that every mango is bad, which would make (C) false while the original statement is still false. So (C) is not necessarily true.
Option (D), "There exists at least one mango in the basket that is not good," is exactly the logical negation of the original statement, so it must be true whenever the original is false.
Step 4: Final Answer:
The statement that is necessarily true is option (D). \[ \boxed{\text{Option (D)}} \]
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