GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 3

Question

Which of the following first-order logic sentence matches closest with the sentence "All students are not equal"?

• A. $\forall x \exists y[\operatorname{student}(x) \wedge \operatorname{student}(y)] \Rightarrow \neg \operatorname{Equal}(x, y)$
• B. $\forall x \forall y[\operatorname{student}(x) \wedge \operatorname{student}(y)] \Rightarrow \neg \operatorname{Equal}(x, y)$
• C. $\forall x \exists y[\operatorname{student}(x) \wedge \operatorname{student}(y) \wedge \neg \operatorname{Equal}(x, y)]$
• D. $\forall x \forall y[\operatorname{student}(x) \wedge \operatorname{student}(y) \wedge \neg \operatorname{Equal}(x, y)]$

   

Answer 1

The sentence "All students are not equal" can be translated into First-Order Logic (FOL) in different ways, but only one option accurately reflects the meaning. Let's analyze each option:

A. ∀𝑥 ∃𝑦[𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑥) ∧ 𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑦)] ⇒ ¬𝐸𝑞𝑢𝑎𝑙(𝑥, 𝑦)

• This translates to "For every student x, there exists another student y such that x is not equal to y."
• This option is incorrect because it doesn't say that all students are different, just that for any student, there exists another who is different.

B. ∀𝑥 ∀𝑦[𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑥) ∧ 𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑦)] ⇒ ¬𝐸𝑞𝑢𝑎𝑙(𝑥, 𝑦)

• This translates to "For every student x and every student y, x is not equal to y."
• This option is too strong because it implies that no student can be equal to themselves, which is not the intended meaning of the original sentence.

C. ∀𝑥 ∃𝑦[𝑠𝑡𝑢დ𝑒𝑛𝑡(𝑥) ∧ 𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑦) ∧ ¬𝐸𝑞𝑢𝑎𝑙(𝑥, 𝑦)]

• This translates to "For every student x, there exists another student y such that x is student and y is student and they are not equal."
• This option is correct. It captures the essence of the original sentence by stating that for any student, there exists another student who is different.

**D. ∀𝑥 ∀𝑦[𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑥) ∧ 𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑦) ∧ ¬𝐸𝑞𝑢𝑎𝑙(𝑥, 𝑦)] **

• This is the same as option B and is therefore too strong and incorrect.

Therefore, the First-Order Logic sentence that closest matches the original sentence "All students are not equal" is C. ∀𝑥 ∃𝑦[𝑠𝑡𝑢𝑑𝑒ន𝑡(𝑥) ∧ 𝑠𝑡𝑢𝑑𝑒𝑛𝑡(𝑦) ∧ ¬𝐸𝑞𝑢𝑎𝑙(𝑥, 𝑦)].

Comments

Option C is correct but your reasoning for why A is incorrect is wrong. You have basically stated option A is the same as option C.

Interpretation of option A would be "For every x there exists a Y such that if X is a student and Y is a student then it implies they are not equal." So it would mean no 2 students are equal which is not followed from the given statement.

Answer 2

4

For all students X &  Y ,if X is a student  & Y is student, then  X  is not equal to Y

Same as no two students  are equal

So all students are not equal

Student (x): true if & only if X is true

Equal(X,y): predicte is true if & only if X any Y are equal

Conclusion: equal (X,y) are not equal

Comments

Bro can you tell which subject question is this and from which book to study it.

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Propositional logic.

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I don't agree @shivakundank.

"All students are not equal" is not equivalent to saying "No two students  are equal".

"All students are not equal" is more close to the statement - "Some students are equal but not all" or "Some student are equal and some are not." or "All the students taken together are not equal"

I that case option C makes more sense.

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@6rivu Depends on the domain of variable x and y, say the domain is everyone at school (including faculty members) then the expression in C and D will become false even if in reality there exists a non equal pair of student as student(some faculty member) is false.

We need a way to ignore all non students members, B should be correct IMO