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Let's analyze the problem and translate the logical statements into their equivalent English sentences.

1. **First statement:**  
   \[
   \forall y \exists x \, T(x, y)
   \]
   This means: "For every course \( y \), there exists at least one student \( x \) such that \( T(x, y) \) is true," i.e., "Every course is being taken by at least one student."  
   **Equivalent to:** (2) Every course is being taken by at least one student.

2. **Second statement:**  
   \[
   \exists x \forall y \, T(x, y)
   \]
   This means: "There exists at least one student \( x \) such that for every course \( y \), \( T(x, y) \) is true," i.e., "Some student is taking every course."  
   **Equivalent to:** (3) Some student is taking every course.

3. **Third statement:**  
   \[
   \neg (\exists x \exists y \, T(x, y))
   \]
   This means: "It is not true that there exists a student \( x \) and a course \( y \) such that \( T(x, y) \) is true," i.e., "No student is taking any course."  
   **Equivalent to:** (1) No student is taking any course.

4. **Fourth statement:**  
   \[
   \neg (\forall x (\forall y \neg T(x, y)))
   \]
   This means: "It is not true that for every student \( x \), they are not taking any course \( y \)," i.e., "Some students are taking no courses."  
   **Equivalent to:** (4) Some students are taking no courses.

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### Final answer mapping:
- (1) corresponds to the third statement: \(\neg (\exists x \exists y \, T(x, y))\)
- (2) corresponds to the first statement: \(\forall y \exists x \, T(x, y)\)
- (3) corresponds to the second statement: \(\exists x \forall y \, T(x, y)\)
- (4) corresponds to the fourth statement: \(\neg (\forall x (\forall y \neg T(x, y)))\)

Would you like further clarification on any part of this problem? 

Here are five related questions to deepen your understanding:
1. What is the difference between "for all" and "there exists" in logic?
2. How do negations affect quantified statements?
3. Can you give another example of a quantified statement involving two variables?
4. How would you translate a logical statement that means "Some courses are not taken by any students"?
5. How would you express the idea that "Every student is taking at least one course" using logical symbols?

**Tip:** Always pay close attention to the order of quantifiers in statements. Reversing them can lead to a completely different meaning!

Suppose the variable x represents students, y represents courses, and T(x,y) means 'x is taking y'. Based on the given quantifier statements, match the statements with the equivalent English sentences.

Learn how to translate quantified logical statements into their equivalent English sentences. This problem covers key logic concepts, including the use of quantifiers (∃ and ∀) and negations. Suitable for students in Grades 11-12.

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