Similar questions in GATE 2021 and different answer keys | Answer Key Challenge GATE CSE 2021 Set 1 | GA Question: 9

Given below are two statements 1 and 2, and two conclusions I and II

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*Statement 1: All bacteria are microorganisms.

*Statement 2: All pathogens are microorganisms.

*Conclusion I: Some pathogens are bacteria.

*Conclusion II: All pathogens are not bacteria.

Based on the above statements and conclusions, which one of the following options is logically CORRECT?

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*(A) Only conclusion I is correct

*(B) Only conclusion II is correct

*(C) Either conclusion I or II is correct

*(D) Neither conclusion I nor II is correct


My attempt:
I have to find counter example,

Case-1 is counter example of Conclusion II, because of all pathogens are bacteria in this diagram.
Conclusion I would be true in this case, because all so also some.
(So, Conclusion I is true and II is false.).
Case-2 is counter example of Conclusion I, because no pathogens are bacteria.
Conclusion II would be true in this case, because all so also some.
(So, Conclusion II is false and II is true.).
Case-3: Conclusion I is true, and Conclusion II is false as some pathogens are already bacteria.
Case-4: Conclusion I is true, and Conclusion II is false as some pathogens are already bacteria.
So, I found counter example for both Conclusion. Hence, option (D) is true (Official Key given by GATE 2021.).
I've read this method for finding counter examples for such questions.

Doubt here, is as both given Conclusions are negation of each other. That is,

∃x(p(x) ∧ b(x)) = ¬∀x (p(x) → ¬b(x))
∀x (p(x) → ¬b(x)) = ¬∃x(p(x) ∧ b(x))

Peoples are debating this question and saying that the correct answer should be option (C) Either conclusion I or II is correct.

A exact similar question appeared in the GATE EC branch, which has same above logic.
GATE EC 2021 | GA Question: 6 :
Given below are two statements and two conclusions.

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*Statement 1: All purple are green.


*Statement 2: All black are green.


*Conclusion I: Some black are purple.


*Conclusion II: No black is purple.
Based on the above statements and conclusions, which one of the following options is logically CORRECT?

*

*(A) Only conclusion I is correct

*(B) Only conclusion II is correct

*(C) Either conclusion I or II is correct

*(D) Both conclusion I and II are correct

Here, as above my attempt, we can find counter examples for the both the statements, so both (D) Both conclusion I and II are correct.
But, this time official key is given (C) Either conclusion I or II is correct.

Note these are two different branches of Engineering (Computer Science and Engineering, Electronics and Communication Engineering). Professors who designed these questions for GATE 2021, should not be same person (but, could be as these questions are from General Aptitude which is common (but not same questions) in all branches of GATE Papers).

My Question

What is correct approach to solve such questions and which answer keys are correct which are wrong ? If both keys are correct respective their questions then what is possible difference between these questions ?

Thank you,

Update :
Final answer keys given by GATE official :

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*For first question (GATE CSE 2021 Set 1 | GA Question: 9) : Both
(C) or (D)

*For second question (GATE EC 2021 | GA Question: 6) : Only (C)

Now, why they have given option (D) as answer for the first question
!?

 A: You're correct in that both questions have exactly the same idea, and in both cases the solution is "Either I or II is correct". We're using the law of the excluded middle, so either $P$ is true or $\neg P$ is. The official answer for the first question seems to be mistaken.
You'll notice that in your attempt, in all your cases $C$ was still true, which should indicate to you that $C$ could be valid, and you shouldn't automatically jump to $D$. With your cases you could confirm $A$ and $B$ were false. But you didn't prove $C$ was false. In this case we can prove $C$ is true with the law I mentioned earlier.
A: Case 1
Statement 1: All B are M.
Statement 2: All P are M.
Conclusion I: Some P are B.
Conclusion II: All P are not B.
Counterexample to I): All Triangles are Plane figures. All Quadrilateral are Plane figures.
Therefore: Some Quadrilateral are Triangles. The conclusion is False.
Counterexample to II): All multiples of 2 are integers. All multiple of 3 are integers.
Therefore: All multiple of 3 is not a multiple of 2. The conclusion is False (consider 6).

We are speaking of logical (or: deductive) validity: an argument is valid by virtue of the "logical form".
The "factual" truth of the purported conclusion does not mean that the argument is formally valid.

What happens with Case 2 ?
We have to note that option (D) is different from the previous one. It reads "Both conclusion I and II are correct."
But I and II are contradictory: thus, they cannot be both true.
Thus, if (C) is the correct answer, the issue is: I or II? "Either" is usually used in logic in the inclusive form. But we have already noted that they cannot be both true.
Thus, "Either I or II" amounts to choosing between (A) and (B).
