Determine the truth value of the following propositional forms given that p is false, q is true, and r is true.

Let's determine the truth values of some common propositional forms given the truth values: \( p \) is false, \( q \) is true, and \( r \) is true.

### Propositional Forms

1. **Conjunction (\( p \land q \))**: The conjunction \( p \land q \) is true if both \( p \) and \( q \) are true.
   - Given: \( p \) is false, \( q \) is true.
   - \( p \land q = \text{false} \land \text{true} = \text{false} \)

2. **Disjunction (\( p \lor q \))**: The disjunction \( p \lor q \) is true if at least one of \( p \) or \( q \) is true.
   - Given: \( p \) is false, \( q \) is true.
   - \( p \lor q = \text{false} \lor \text{true} = \text{true} \)

3. **Implication (\( p \rightarrow q \))**: The implication \( p \rightarrow q \) is true unless \( p \) is true and \( q \) is false.
   - Given: \( p \) is false, \( q \) is true.
   - \( p \rightarrow q = \text{false} \rightarrow \text{true} = \text{true} \)

4. **Biconditional (\( q \leftrightarrow r \))**: The biconditional \( q \leftrightarrow r \) is true if both \( q \) and \( r \) have the same truth value.
   - Given: \( q \) is true, \( r \) is true.
   - \( q \leftrightarrow r = \text{true} \leftrightarrow \text{true} = \text{true} \)

5. **Negation (\( \neg p \))**: The negation \( \neg p \) is true if \( p \) is false.
   - Given: \( p \) is false.
   - \( \neg p = \neg \text{false} = \text{true} \)

### Summary of Truth Values:
- \( p \land q \) : **False**
- \( p \lor q \) : **True**
- \( p \rightarrow q \) : **True**
- \( q \leftrightarrow r \) : **True**
- \( \neg p \) : **True**

Would you like more details on any of these, or do you have any other questions?

Here are 5 related questions you might find useful:
1. What is the truth value of \( \neg q \) given that \( q \) is true?
2. How does the truth table for a conjunction work?
3. What is the truth value of \( (p \lor r) \rightarrow q \)?
4. Can you explain how to construct the truth table for \( p \rightarrow q \)?
5. What is the truth value of \( p \land (q \lor r) \)?

**Tip:** Always double-check the truth table of each logical operation when working with propositional forms.

Explore the truth values of various propositional forms with specific truth values: p is false, q is true, and r is true. Learn about conjunction, disjunction, implication, biconditional, and negation in propositional logic.

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