 Let p be the proposition “You get an A on the final exam,” let q be the proposition “You did every exercise

in this book,” and let r be the proposition “You get an A in this class.” Write these propositions using 𝑝𝑝,

𝑞𝑞, and 𝑟𝑟 and logical connectives.

a) You get an A in this class but did not do every exercise in this book.

b) You get an A on the final, do every exercise in this book, and get an A in this class.

c) To get an A in this class, it is necessary for you to get an A on the final.

d) You get an A on the final but did not do every exercise in this book; nevertheless, you get an A in

this class.

e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this

class.

Let's express each of these propositions using the symbols \( p \) (you get an A on the final exam), \( q \) (you did every exercise in this book), and \( r \) (you get an A in this class), along with logical connectives.

### a) You get an A in this class but did not do every exercise in this book.
This can be expressed as:
\[ r \land \neg q \]

### b) You get an A on the final, do every exercise in this book, and get an A in this class.
This can be expressed as:
\[ p \land q \land r \]

### c) To get an A in this class, it is necessary for you to get an A on the final.
This can be expressed as:
\[ r \rightarrow p \]

### d) You get an A on the final but did not do every exercise in this book; nevertheless, you get an A in this class.
This can be expressed as:
\[ p \land \neg q \land r \]

### e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class.
This can be expressed as:
\[ (p \land q) \rightarrow r \]

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Do you want details on any of these logical expressions, or do you have any questions? 

Here are five questions that extend these concepts:
1. How do we represent the logical expression for "If you did every exercise in the book, then you get an A on the final"?
2. What is the contrapositive of the expression \( r \rightarrow p \)?
3. How would you express "Getting an A in this class is sufficient for getting an A on the final and doing every exercise"?
4. How can we represent "You did not get an A on the final and did every exercise in this book"?
5. What is the converse of the statement \( (p \land q) \rightarrow r \)?

**Tip:** When working with logical connectives, understanding the difference between "necessary" and "sufficient" conditions is crucial for accurate translation into logical expressions.

Explore key logical expressions using propositional logic, including statements involving propositions p, q, and r. Learn about conjunctions, disjunctions, implications, and logical equivalence. Suitable for advanced high school students and beyond.

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