Students complete a worksheet on conditional statements, identifying truth values and classifying statements as conditional, converse, contrapositive, or bi-conditional.

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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 with its related forms.

#### a) Inverse:
> If a number is not a rational number, then it is not a whole number.

- Truth Value: True
- Reason: The inverse is logically equivalent to the original only if both are true or false. But here, the original statement is false, so we need to check.
- However, the inverse is not necessarily false just because the original is false.
- But let’s think: There are numbers that are not rational (like √2), and they are not whole numbers → So the inverse holds in this case.
- But note: The inverse does not have to be true just because the original is false.
- Actually, the inverse is logically independent of the original.
- But since the original is false, and the inverse may or may not be true — let's re-evaluate.

Wait! Let's first assess the original statement.

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Evaluate Original Statement:

> If a number is a rational number, then it is also a whole number.

This is false.

- Example: $ \frac{1}{2} $ is a rational number, but not a whole number.
- So the implication fails → False

So the truth value of the original statement is False.

Now, let’s evaluate each form.

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#### a) Inverse:
> If a number is not rational, then it is not a whole number.

- This says: Non-rational ⇒ Not whole
- Example: $ \sqrt{2} $ is not rational, and it's not a whole number → OK
- But what about non-rational numbers that are whole numbers? None exist — all whole numbers are rational.
- So: All whole numbers are rational ⇒ So if a number is not rational, it cannot be a whole number.
- Therefore, the inverse is true.

✔ Truth Value: True

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#### b) Converse:
> If a number is a whole number, then it is a rational number.

- Whole numbers: 0, 1, 2, 3, ...
- All whole numbers can be written as fractions: $ \frac{0}{1}, \frac{1}{1}, \frac{2}{1}, \dots $
- So yes, all whole numbers are rational.

✔ Truth Value: True

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#### c) Contrapositive:
> If a number is not a whole number, then it is not a rational number.

- Is this true?
- Counterexample: $ \frac{1}{2} $ is not a whole number, but is rational.
- So the contrapositive is false.

✘ Truth Value: False

Note: Contrapositive should have the same truth value as the original.
- Original was false → Contrapositive must also be false → Correct.

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#### d) Biconditional:
> A number is a rational number if and only if it is a whole number.

This means: Rational ⇔ Whole

But this is false, because:
- Some rationals are not whole (e.g., 1/2)
- Some whole numbers are rational (true), but not all rationals are whole

So the biconditional is false.

✘ Truth Value: False

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✔ Summary for Problem 1:

| Form | Statement | Truth Value |
|------|----------|-------------|
| Original | If rational → whole | False |
| Inverse | If not rational → not whole | True |
| Converse | If whole → rational | True |
| Contrapositive | If not whole → not rational | False |
| Biconditional | Rational ↔ whole | False |

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Problem 2: Match Logical Forms

Given:
- $ p $: You have a library card
- $ q $: You can check out books

We are to match symbolic expressions with English statements and identify the type.

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#### a) $ p \rightarrow q $

> "If you have a library card, then you can check out books."

- This is the conditional (if-then).
- Classification: Conditional

✔ Answer: Conditional

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#### b) $ \neg q \rightarrow \neg p $

> "If you cannot check out books, then you do not have a library card."

- This is the contrapositive of $ p \rightarrow q $
- Because contrapositive is $ \neg q \rightarrow \neg p $
- Classification: Contrapositive

✔ Answer: Contrapositive

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#### c) $ q \rightarrow p $

> "If you can check out books, then you have a library card."

- This is the converse of $ p \rightarrow q $
- Classification: Converse

✔ Answer: Converse

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#### d) $ \neg p \rightarrow \neg q $

> "If you don't have a library card, then you cannot check out books."

- This is the inverse of $ p \rightarrow q $
- Classification: Inverse

✔ Answer: Inverse

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#### e) $ p \leftrightarrow q $

> "You have a library card if and only if you can check out books."

- This is the biconditional
- Classification: Biconditional

✔ Answer: Biconditional

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✔ Final Answers for Problem 2:

| Symbolic | English Statement | Type |
|---------|-------------------|------|
| a) $ p \rightarrow q $ | If you have a library card, then you can check out books | Conditional |
| b) $ \neg q \rightarrow \neg p $ | If you cannot check out books, then you do not have a library card | Contrapositive |
| c) $ q \rightarrow p $ | If you can check out books, then you have a library card | Converse |
| d) $ \neg p \rightarrow \neg q $ | If you don't have a library card, then you cannot check out books | Inverse |
| e) $ p \leftrightarrow q $ | You have a library card if and only if you can check out books | Biconditional |

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🟩 Final Notes:

- The original conditional ("If rational → whole") is false.
- The converse and inverse are true, but that doesn’t make the original true.
- Only conditional and contrapositive have the same truth value.
- The biconditional requires both directions to be true — which isn’t the case here.

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