Use truth tables to establish the truth of each statement A conditional statement is not logically equivalent to its inverse.

To establish the truth that a conditional statement is not logically equivalent to its inverse using truth tables, we need to:

1. Define the **conditional statement** (if \( p \), then \( q \)) as \( p \rightarrow q \).
2. Define the **inverse** of the conditional statement (if **not** \( p \), then **not** \( q \)) as \( \neg p \rightarrow \neg q \).

### Step-by-Step Process:

#### 1. Define the symbols:
- \( p \): Proposition 1.
- \( q \): Proposition 2.
- \( p \rightarrow q \): Conditional statement.
- \( \neg p \rightarrow \neg q \): Inverse of the conditional statement.

#### 2. Construct the truth table:
We’ll build a truth table to evaluate the truth values of both \( p \rightarrow q \) and \( \neg p \rightarrow \neg q \) and compare them.

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

#### 3. Analyze the truth table:
- **\( p \rightarrow q \)** (the conditional statement) and **\( \neg p \rightarrow \neg q \)** (the inverse) are **not logically equivalent**, because their truth values differ in the third row. 
- In row 3, \( p \rightarrow q \) is true, while \( \neg p \rightarrow \neg q \) is false.

This difference confirms that a conditional statement is not logically equivalent to its inverse.

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Would you like further clarification or details on any step? Here are 5 related questions to deepen your understanding:

1. How does the converse of a conditional statement differ from the inverse?
2. Can you find an example of a conditional statement and its inverse in real-life situations?
3. How is the contrapositive of a conditional statement related to its truth value?
4. Why is it important to understand the difference between logical equivalence and non-equivalence?
5. How would the truth table change for the biconditional statement \( p \leftrightarrow q \)?

**Tip**: The **contrapositive** of a conditional statement is always logically equivalent to the original conditional.

Use truth tables to establish the truth of each statement: A conditional statement is not logically equivalent to its inverse.

Learn how to use truth tables to prove that a conditional statement (p → q) is not logically equivalent to its inverse (¬p → ¬q). This problem covers key concepts in propositional logic, including conditional and inverse statements, and explores logical equivalence through truth table analysis. Suitable for students in Grades 10-12.

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