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Let's break down and solve the problem step by step.

### Step 1: Define Simple Statements
Let's denote the following simple statements from the problem:
- \( p \): "Butch bought the required calculator for the course."
- \( q \): "Butch passed the course."

### Step 2: Translate the Statements into Symbolic Logic
1. **First Statement (If Butch did not buy the calculator, then she cannot pass the course)**:
   - This can be written as a conditional statement: 
   - "If not \( p \), then not \( q \)."
   - Symbolically: \( \neg p \rightarrow \neg q \)

2. **Second Statement (Butch bought the required calculator for the course and she passed the course)**:
   - This is a conjunction of two statements:
   - "Butch bought the calculator" (which is \( p \)) and "she passed the course" (which is \( q \)).
   - Symbolically: \( p \land q \)

### Step 3: Truth Table
Let's construct a truth table for both statements and compare them.

| \( p \) | \( q \) | \( \neg p \) | \( \neg q \) | \( \neg p \rightarrow \neg q \) | \( p \land q \) |
|--------|--------|------------|------------|---------------------------|-------------|
| T      | T      | F          | F          | T                         | T           |
| T      | F      | F          | T          | T                         | F           |
| F      | T      | T          | F          | F                         | F           |
| F      | F      | T          | T          | T                         | F           |

### Step 4: Explanation of Truth Table
- **For the first statement** \( \neg p \rightarrow \neg q \):
   - The conditional is only false when the antecedent (\( \neg p \)) is true, and the consequent (\( \neg q \)) is false. In this case, the truth table shows the statement is false when \( p = F \) and \( q = T \), which matches the logical rule for implications.
- **For the second statement** \( p \land q \):
   - The conjunction is only true when both \( p \) and \( q \) are true, which occurs only in the first row of the truth table.

### Step 5: Are the Statements Logically Equivalent?
- The statements are **not logically equivalent** because their truth values differ across the rows in the truth table. Specifically, the first statement \( \neg p \rightarrow \neg q \) is true in the third row, while the second statement \( p \land q \) is false in that same row.

### Conclusion
The two statements are **not logically equivalent** as their truth tables differ.

Would you like to dive deeper into any part of this explanation or have further questions?

#### Related Questions:
1. What does it mean for two statements to be logically equivalent?
2. How do you construct a truth table for more complex logical statements?
3. What is the significance of a truth table in determining logical equivalence?
4. Can conditional statements (\( \rightarrow \)) be written in different logical forms?
5. How does negation affect the truth values in logical statements?

#### Tip:
In logical reasoning, always start by defining the simple components of a statement clearly before translating them into symbolic form to avoid confusion.

Write the following two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and explanation.

Explore how to express logical statements in symbolic form and determine their equivalence using truth tables. This problem involves propositional logic, covering implications, conjunctions, and negation.

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