Do the following statement and its converse form a biconditional that is true?

If a number is a multiple of 9, then it is divisible by 3.



To determine whether the given statement and its converse form a true biconditional, we need to analyze both the original statement and its converse.

### Original Statement:
- **Statement:** "If a number is a multiple of 9, then it is divisible by 3."
- **Truth:** This statement is true. Any multiple of 9 is divisible by 3 because 9 is itself a multiple of 3.

### Converse Statement:
- **Converse:** "If a number is divisible by 3, then it is a multiple of 9."
- **Truth:** This statement is **false**. Not all numbers divisible by 3 are multiples of 9. For example, 6 is divisible by 3 but is not a multiple of 9.

### Biconditional:
A biconditional statement is true only if both the original statement and its converse are true. Since the converse is false, the biconditional formed by these two statements is not true.

**Conclusion:** The original statement and its converse do not form a biconditional that is true.

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. What is the definition of a biconditional statement in logic?
2. Can you give an example of a true biconditional statement?
3. What is the difference between the converse, inverse, and contrapositive of a statement?
4. How can we determine if a biconditional statement is true?
5. What other pairs of statements do not form a true biconditional?

**Tip:** To form a true biconditional, both the original statement and its converse must be true. This means the conditions must be necessary and sufficient for each other.

Explore the concept of biconditional statements in logic with an example analysis. Learn why the statement 'If a number is a multiple of 9, then it is divisible by 3' and its converse do not form a true biconditional. Gain insights into logic reasoning and related concepts.

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