What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some studenta)∀(x)[teacher(x)→∃(y)[student(y)→likes(y,x)]]b)∀(x)[teacher(x)→∃(y)[student(y)&and;likes(y,x)]]c)∃(y)∀(x)[teacher(x)→[student(y)&and;likes(y,x)]]d)∀(x)[teacher(x)&and;∃(y)[student(y)→likes(y,x)]]Correct answer is option 'B'. Can you explain this answer?

Question

What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some studenta)&forall;(x)[teacher(x)&rarr;&exist;(y)[student(y)&rarr;likes(y,x)]]b)&forall;(x)[teacher(x)&rarr;&exist;(y)[student(y)&and;likes(y,x)]]c)&exist;(y)&forall;(x)[teacher(x)&rarr;[student(y)&and;likes(y,x)]]d)&forall;(x)[teacher(x)&and;&exist;(y)[student(y)&rarr;likes(y,x)]]Correct answer is option 'B'. Can you explain this answer?

Answer

Understanding the Statement
The statement 'Every teacher is liked by some student' can be broken down into two key components:
- Universal quantification: 'Every teacher' indicates that we are talking about all individuals in the category of teachers.
- Existential quantification: 'some student' signifies that there is at least one student for each teacher who likes them.
Analyzing the Options
Let's examine the provided options to identify the correct logical representation:
- Option A: This option uses an implication (if-then) structure incorrectly. It suggests that if someone is a teacher, then there exists a student who is liked by the teacher, which is not the intended meaning.
- Option B: This option states that for every individual x, if x is a teacher, then there exists some individual y such that y is a student and y likes x. This is a correct translation because it asserts that for each teacher, there is at least one student who likes them.
- Option C: This option incorrectly claims that there exists a particular student y such that for all teachers x, that student y likes every teacher. This does not capture the original statement’s intention.
- Option D: This option suggests that every teacher must be liked by some student, but the structure is incorrectly phrased, leading to a misunderstanding of the relationship.
Conclusion
The correct answer is:
- Option B: It correctly reflects the meaning of the original statement by ensuring that for every teacher, there exists at least one student who likes them. This captures both the universal and existential aspects of the statement accurately.

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