Question

Which of the given logical forms correspond to the following sentence where \( x \in \) student, \( y \) teacher and \( A = \) admires?
Every student admires a teacher.

• A. \( \forall x \exists y \, A(x,y) \)
• B. \( \forall y \exists x \, A(y,x) \)
• C. \( \exists x \forall y \, A(y,x) \)
• D. \( \exists y \forall x \, A(x,y) \)

Hint:

In logical forms, use the universal quantifier \( \forall \) for "every" and the existential quantifier \( \exists \) for "there exists."

Answer 1

The Correct Option is A, D

Step 1: Understanding the sentence.
The sentence "Every student admires a teacher" implies that for each student, there exists at least one teacher whom they admire. This requires an existential quantifier for the teacher and a universal quantifier for the student.
Step 2: Analyzing the options.
- (A) Correct. \( \forall x \exists y \, A(x,y) \) means that for every student \( x \), there exists a teacher \( y \) such that \( x \) admires \( y \), which correctly reflects the meaning of the sentence.
- (B) Incorrect. This statement suggests that every teacher is admired by some student, which reverses the intended meaning.
- (C) Incorrect. \( \exists x \forall y \, A(y,x) \) suggests that there exists a student who admires every teacher, which is not the same as the original sentence.
- (D) Correct. \( \exists y \forall x \, A(x,y) \) means that there exists a teacher \( y \) who is admired by every student \( x \), which also matches the logical structure of the sentence.
Step 3: Conclusion.
Thus, the correct answers are (A) and (D).