GATE CE 2026 Set 2 — Question 4
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General Aptitude → Analytical Aptitude → Deduction & Induction
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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?
If the above statement is false, then which one of the following statements is necessarily true?
Correct answer
(D) There exists at least one mango in the basket that is not good.
Solution
The original statement is: "All the mangoes in the basket are good."If this statement is false, it means that it is NOT the case that all mangoes are good. The negation of "All A are B" is "Some A are not B" or "There exists at least one A that is not B."Let's apply this to the given statement:
If "All the mangoes in the basket are good" is false, then it must be true that "There exists at least one mango in the basket that is not good."Let's evaluate the options:
(A) "All the mangoes in the basket are not good." This is a stronger statement than the negation. If the original statement is false, it doesn't necessarily mean all mangoes are bad; it just means not all are good. Some could still be good.
(B) "No mango in the basket is good." This is equivalent to "All mangoes are not good," which is the same as option (A) and is not necessarily true.
(C) "In the basket, some of the mangoes are good and some are not good." This is one possibility if the original statement is false, but it's not necessarily true. It's possible that only one mango is not good, and all others are good. Or it's possible that all mangoes are not good. This statement implies a mix, which isn't guaranteed.
(D) "There exists at least one mango in the basket that is not good." This is the direct and necessary negation of the original statement. If it's false that all mangoes are good, then it must be true that at least one mango is not good.Therefore, option (D) is the necessarily true statement.
If "All the mangoes in the basket are good" is false, then it must be true that "There exists at least one mango in the basket that is not good."Let's evaluate the options:
(A) "All the mangoes in the basket are not good." This is a stronger statement than the negation. If the original statement is false, it doesn't necessarily mean all mangoes are bad; it just means not all are good. Some could still be good.
(B) "No mango in the basket is good." This is equivalent to "All mangoes are not good," which is the same as option (A) and is not necessarily true.
(C) "In the basket, some of the mangoes are good and some are not good." This is one possibility if the original statement is false, but it's not necessarily true. It's possible that only one mango is not good, and all others are good. Or it's possible that all mangoes are not good. This statement implies a mix, which isn't guaranteed.
(D) "There exists at least one mango in the basket that is not good." This is the direct and necessary negation of the original statement. If it's false that all mangoes are good, then it must be true that at least one mango is not good.Therefore, option (D) is the necessarily true statement.
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