Define converse and inverse and contropositive

In logic, converse, inverse, and contrapositive refer to different forms of conditional statements derived from a given statement. Suppose we have a conditional statement:

- **Original Statement (Conditional)**: "If \( p \), then \( q \)" (symbolically, \( p \rightarrow q \)).

Now, let’s define each form based on this conditional statement:

### 1. **Converse**:
   The **converse** of a statement switches the hypothesis and conclusion. So, for the statement \( p \rightarrow q \), the converse is:
   
   - **Converse**: "If \( q \), then \( p \)" (symbolically, \( q \rightarrow p \)).

### 2. **Inverse**:
   The **inverse** of a statement negates both the hypothesis and the conclusion. So, for the statement \( p \rightarrow q \), the inverse is:
   
   - **Inverse**: "If not \( p \), then not \( q \)" (symbolically, \( \neg p \rightarrow \neg q \)).

### 3. **Contrapositive**:
   The **contrapositive** is a combination of both the converse and the inverse: it negates and switches both the hypothesis and the conclusion. So, for the statement \( p \rightarrow q \), the contrapositive is:
   
   - **Contrapositive**: "If not \( q \), then not \( p \)" (symbolically, \( \neg q \rightarrow \neg p \)).

### Truth Relationships:
- A conditional statement and its contrapositive are logically **equivalent** (they always have the same truth value).
- Similarly, the converse and the inverse of a statement are logically equivalent to each other, but not necessarily to the original statement.

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Would you like further details on how to apply these concepts or examples of them in use?

### Related Questions:
1. What is the difference between logical equivalence and truth tables?
2. Can a converse of a true statement be false?
3. How do you prove the contrapositive of a conditional statement?
4. How do these logic forms apply in mathematical proofs?
5. Can the inverse of a statement ever be equivalent to the original?

### Tip:
When trying to disprove a statement, often it's easier to prove its contrapositive rather than the original.

Define converse and inverse and contrapositive

Explore the definitions and relationships of converse, inverse, and contrapositive in logic. This topic delves into conditional statements and their various forms, including important concepts such as logical equivalence. Suitable for students in Grades 9-12.

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