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### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Let p, q, and r be the propositions p : You get an A on the final exam. q : You do every exercise in this book. r : You get an A in this class. Write these propositions using p, q, and r and logical connectives (including negations). a)You get an A in this class, but you do not do every exercise in this book.b) You get an A on the final, you do every exercise in this book, and you 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 you don’t 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. f ) You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final.

a) $r\wedge ¬q$

b) $p\wedge q\wedge r$

c) $r\to p$

d) $p\wedge ¬q\wedge r$

e) $\left(p\wedge q\right)\to r$

f) $r↔\left(p\vee q\right)$

See the step by step solution

## Step1: Definition of proposition

Proposition:
1. It is a declarative statement which can be either true or false. 2. It cannot be both true and false simultaneously

## Step 2: Proposition using p, q and r for a)

p: You get an A on the final exam.

q: You do every exercise in this book.

r: You get an A in this class.

You get an A in this class, but you do not do every exercise in this book. The above given statement means $r\wedge ¬q$.

## Step 3:Proposition using p and q for b)

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

The given statement means $p\wedge q\wedge r$

## Step 4: Proposition using p, q and r for c)

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

The above given statement means $r\to p$

## Step 5: Proposition using p, q and r  for d)

You get an A on the final, but you don’t do every exercise in this book; never the less, you get an A in this class

The given statement means $p\wedge ¬q\wedge r$.

## Step 6: Proposition using p, q and r for e)

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

The given statement means $\left(p\wedge q\right)\to r$.

## Step 7: Proposition using p, q and r for f)

You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final.

The given statement means $r↔\left(p\vee q\right)$