https://file.aiquickdraw.com/mathbot/file/1736637747569y37cznbv.jpg

Let's break down the problem you've provided:

The conditional statement is:  
**"If you study well, then you will pass the exam."**

### Part (a): Write the converse, the inverse, and the contrapositive.

1. **Converse**:  
   This is formed by switching the hypothesis and conclusion.  
   **Statement**: "If you pass the exam, then you studied well."

2. **Inverse**:  
   This is formed by negating both the hypothesis and conclusion of the original statement.  
   **Statement**: "If you do not study well, then you will not pass the exam."

3. **Contrapositive**:  
   This is formed by both switching and negating the hypothesis and conclusion of the original statement.  
   **Statement**: "If you do not pass the exam, then you did not study well."

---

### Part (b): Verify each statement or disprove it with a counterexample.

- **Original Statement**:  
  "If you study well, then you will pass the exam."  
  - This may be true, assuming that studying well generally leads to passing an exam. However, this could be disproved if there exists a situation where someone studied well but did not pass, perhaps due to unforeseen circumstances like illness or misinterpreting the exam content.

- **Converse**:  
  "If you pass the exam, then you studied well."  
  - This is not necessarily true. There could be cases where someone passes the exam without studying well, possibly by guessing or having prior knowledge of the material. A counterexample would be someone who passes but did not study much.

- **Inverse**:  
  "If you do not study well, then you will not pass the exam."  
  - This could be generally true but not always. Some people may pass exams without studying well, due to luck or other factors. A counterexample would be someone who did not study well but still passed.

- **Contrapositive**:  
  "If you do not pass the exam, then you did not study well."  
  - This statement logically holds if the original statement is true. If someone did not pass the exam, it would imply they did not study well, assuming the original implication is correct.

Would you like to go through any of these in more detail?

---

Here are five related questions to expand your understanding:
1. How do we define a conditional statement in logic?
2. What is the logical equivalence between a statement and its contrapositive?
3. Can the converse of a statement ever have the same truth value as the original statement?
4. How do negations work in forming the inverse of a statement?
5. How do we use counterexamples to disprove logical statements?

**Tip:** In logic, a statement and its contrapositive always share the same truth value, which is a key property when analyzing logical arguments.

Consider this conditional statement: 'If you study well, then you will pass the exam.' a) Write the converse, the inverse, and the contrapositive. b) Verify that each statement is true, or disprove it with a counterexample.

Learn how to analyze a conditional statement by writing its converse, inverse, and contrapositive. Verify each statement's truth with examples. This lesson is ideal for high school students exploring logical reasoning in mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation