Worksheet on conditional statements and logical reasoning in mathematics.

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Problem 1: Conditional Statements and Truth Values

We are given a conditional statement:
> "If a number is a rational number, then it is also a whole number."

Let’s analyze this using logic:

#### Definitions:
- Rational number: A number that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- Whole number: Non-negative integers (0, 1, 2, 3, ...).

So, not all rational numbers are whole numbers (e.g., \( \frac{1}{2} \) is rational but not a whole number). But every whole number *is* a rational number.

Let’s evaluate each part:

1. Original Statement:
> If a number is rational, then it is a whole number.
- Truth Value: False
→ Counterexample: \( \frac{1}{2} \) is rational but not a whole number.

2. Converse:
> If a number is a whole number, then it is a rational number.
- Truth Value: True
→ Every whole number \( n \) can be written as \( \frac{n}{1} \), so it is rational.

3. Contrapositive:
> If a number is not a whole number, then it is not rational.
- Truth Value: False
→ Counterexample: \( \frac{1}{2} \) is not a whole number, but it is rational.

4. Inverse:
> If a number is not rational, then it is not a whole number.
- Truth Value: True
→ If a number isn’t rational, it can't be expressed as a fraction — so it definitely isn't a whole number (since all whole numbers are rational).

> ✔ Note: The contrapositive of a conditional is logically equivalent to the original statement. Since the original is false, the contrapositive is also false.

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Problem 2: Matching Logical Forms

Given:
- \( p \): You have a library card
- \( q \): You can check out books

We match logical forms to English statements.

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#### a) \( p \rightarrow q \)
> "If you have a library card, then you can check out books."
- Classification: Conditional
✔ This is the original conditional statement.

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#### b) \( \neg q \rightarrow \neg p \)
> "If you cannot check out books, then you do not have a library card."
- Classification: Contrapositive
✔ This is the contrapositive of \( p \rightarrow q \). It is logically equivalent to the original.

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#### c) \( q \rightarrow p \)
> "If you can check out books, then you have a library card."
- Classification: Converse
✔ This swaps the hypothesis and conclusion.

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#### d) \( \neg p \rightarrow \neg q \)
> "If you don't have a library card, then you cannot check out books."
- Classification: Inverse
✔ This negates both parts of the original conditional.

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#### e) \( p \leftrightarrow q \)
> "If you have a library card, then you can check out books, and if you can check out books, then you have a library card."
- Classification: Biconditional
✔ This means \( p \) and \( q \) are equivalent.

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✔ Summary Table:

| Logical Form | Statement | Classification |
|---------------------|---------------------------------------------------------------------------|----------------------|
| \( p \rightarrow q \) | If you have a library card, then you can check out books. | Conditional |
| \( \neg q \rightarrow \neg p \) | If you cannot check out books, then you don’t have a library card. | Contrapositive |
| \( q \rightarrow p \) | If you can check out books, then you have a library card. | Converse |
| \( \neg p \rightarrow \neg q \) | If you don’t have a library card, then you can’t check out books. | Inverse |
| \( p \leftrightarrow q \) | You have a library card if and only if you can check out books. | Biconditional |

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Final Notes:

- Conditional: \( p \rightarrow q \)
- Converse: \( q \rightarrow p \)
- Inverse: \( \neg p \rightarrow \neg q \)
- Contrapositive: \( \neg q \rightarrow \neg p \)
- Biconditional: \( p \leftrightarrow q \)

And remember:
- The contrapositive is logically equivalent to the original.
- The converse and inverse are not necessarily true even if the original is.

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