Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people.

All students in your class can solve quadratic equations

To translate the statement "All students in your class can solve quadratic equations" into logical expressions, we will consider two cases: one where the domain consists of **students in your class** and the other where the domain consists of **all people**.

### **Case 1: Domain consists of students in your class**

Let:
- \( S(x) \): \( x \) is a student in your class.
- \( Q(x) \): \( x \) can solve quadratic equations.

1. **Without using \( S(x) \) explicitly** (since the domain is restricted to your class):
   \[
   \forall x \, Q(x)
   \]
   This says "For all \( x \), \( x \) can solve quadratic equations."

2. **Using \( S(x) \)**:
   \[
   \forall x \, (S(x) \rightarrow Q(x))
   \]
   This says "For all \( x \), if \( x \) is a student in your class, then \( x \) can solve quadratic equations."

### **Case 2: Domain consists of all people**

Let:
- \( P(x) \): \( x \) is a person (the domain is all people).
- \( S(x) \): \( x \) is a student in your class.
- \( Q(x) \): \( x \) can solve quadratic equations.

1. **Using \( S(x) \) to specify students in your class**:
   \[
   \forall x \, (S(x) \rightarrow Q(x))
   \]
   This is similar to the second version of Case 1, meaning "For all \( x \), if \( x \) is a student in your class, then \( x \) can solve quadratic equations."

2. **Explicitly stating that all students in your class, among all people, can solve quadratic equations**:
   \[
   \forall x \, (P(x) \land S(x) \rightarrow Q(x))
   \]
   This means "For all \( x \), if \( x \) is a person and \( x \) is a student in your class, then \( x \) can solve quadratic equations."

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Would you like more details on these translations or explanations on specific parts? 

Here are some related questions:
1. How would the logical expression change if some students in the class can't solve quadratic equations?
2. What if the statement was "No students in your class can solve quadratic equations"?
3. How would you express "Some students in your class can solve quadratic equations"?
4. What if the statement was "All students and their friends can solve quadratic equations"?
5. How would the expression change if the problem was generalized to solving "any equation"?

**Tip:** When working with logical expressions, always clarify the domain to avoid ambiguity in your interpretation.

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. All students in your class can solve quadratic equations.

Explore the translation of 'All students in your class can solve quadratic equations' into logical expressions using predicates, quantifiers, and logical connectives. This detailed explanation covers both cases where the domain is restricted to students and when it includes all people. Suitable for Grades 9-12.

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