https://file.aiquickdraw.com/mathbot/file/1726484672221mumlbfw4.jpg

The image shows a logic problem with propositions \( p \), \( q \), and \( r \), defined as:

- \( p \): You have the flu.
- \( q \): You miss the final examination.
- \( r \): You pass the course.

We are asked to express the given logical statements in English. Let’s translate them one by one.

### a) \( p \rightarrow q \)
- **If you have the flu, then you miss the final examination.**

### b) \( q \rightarrow \neg r \)
- **If you miss the final examination, then you do not pass the course.**

### c) \( (p \rightarrow \neg r) \vee (q \rightarrow \neg r) \)
- **Either if you have the flu, then you do not pass the course, or if you miss the final examination, then you do not pass the course.**

### d) \( (p \land q) \vee (\neg q \land r) \)
- **Either you have the flu and miss the final examination, or you do not miss the final examination and pass the course.**

Would you like further details or have any questions on how this translation is done?

Here are some related questions for further understanding:
1. What is the truth table for the implication \( p \rightarrow q \)?
2. How would the negation of \( p \rightarrow q \) be expressed in English?
3. How do we interpret \( \neg p \) in the context of these propositions?
4. How does logical disjunction (OR, \( \vee \)) affect compound statements like in part (c)?
5. What happens if both conditions in part (d) are true?

**Tip:** In logic, \( \rightarrow \) represents an implication, meaning "if...then..." while \( \vee \) is used for "or", and \( \land \) for "and".

Express each of the given logical propositions in English.

This problem involves translating logical propositions with implications, conjunctions, and disjunctions into plain English. Suitable for students learning propositional logic concepts like 'if-then' statements and compound logical operations.

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