'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
- 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.
The Correct Option is D
Solution and Explanation
The statement "All the mangoes in the basket are good" is a universal statement of the form "All A are B". We are told this statement is false, and we need to find what must necessarily be true as a result.
Step 2: Key Formula or Approach.
In logic, the negation of "All A are B" is not "No A are B" and not "All A are not B". The negation of a universal statement is an existential statement: "There exists at least one A that is not B". This is because "All A are B" fails as soon as even a single A fails to be B; it does not require every A to fail.
Step 3: Detailed Explanation.
Here, A is "mango in the basket" and B is "good". If "All the mangoes in the basket are good" is false, it only means this claim is not universally true for every mango. It says nothing about how many mangoes are not good, only that at least one mango fails to be good. So the necessarily true statement is: "There exists at least one mango in the basket that is not good."
Now check why the other options are not necessarily true:
Option (A), "All the mangoes in the basket are not good," would mean every single mango is bad, which is a much stronger claim than the negation requires; the false original statement does not force every mango to be bad.
Option (B), "No mango in the basket is good," says the same thing as option (A) in different words, and is also too strong.
Option (C), "In the basket, some of the mangoes are good and some are not good," assumes there is a mix of good and bad mangoes, but the false statement is also consistent with every mango being bad (in which case there are no good mangoes at all), so option (C) is not guaranteed.
Step 4: Final Answer.
The only statement that must be true, in every case where the original statement is false, is that there exists at least one mango in the basket that is not good.
\[ \boxed{\text{Option (D)}} \]
Top GATE MT General Aptitude Questions
Despite his initial hesitation, Rehman’s ____________ to contribute to the success of the project never wavered.
Select the most appropriate option to complete the above sentence.Bird : Nest :: Bee : __________
Select the correct option to complete the analogy.- If \( P e^x = Q e^{-x} \) for all real values of \( x \), which one of the following statements is true?
The paper as shown in the figure is folded to make a cube where each square corresponds to a particular face of the cube. Which one of the following options correctly represents the cube? Note: The figures shown are representative.
- Let \( p_1 \) and \( p_2 \) denote two arbitrary prime numbers. Which one of the following statements is correct for all values of \( p_1 \) and \( p_2 \)?
Top GATE MT Analytical Aptitude - Seating Arrangement Questions
- Consider the following statements about four numbers:
(S1) The average of the four numbers is 25
(S2) Each number is at most 40
(S3) Each number is at least 20
Choose the option that is necessarily correct. - The Roman senator Meritorius, his brother, his son, and his daughter have varying oratory skill levels. They are seated in rows and columns as shown in the figure with exactly one person sitting in each box. It is known that
(i) Meritorius' daughter and his brother are seated in the same column.
(ii) His son is seated diagonally across the sibling of the worst orator.
(iii) The best and worst orators are seated in the same row.
Who is the best orator?
Column 1 Column 2 Row 1 Row 2
Top GATE MT Questions
Despite his initial hesitation, Rehman’s ____________ to contribute to the success of the project never wavered.
Select the most appropriate option to complete the above sentence.Bird : Nest :: Bee : __________
Select the correct option to complete the analogy.- If \( P e^x = Q e^{-x} \) for all real values of \( x \), which one of the following statements is true?
The paper as shown in the figure is folded to make a cube where each square corresponds to a particular face of the cube. Which one of the following options correctly represents the cube? Note: The figures shown are representative.
- Let \( p_1 \) and \( p_2 \) denote two arbitrary prime numbers. Which one of the following statements is correct for all values of \( p_1 \) and \( p_2 \)?