2024年4月5日发(作者：)

Logic 1

1. 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 q,q and r and logical connectives.

a) you get an A in this class, but you don't 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.

2. Inhabitants of the island Smullyan are either Knights or Knaves. Knights always tell the

truth while knaves tell lies. You encounter two people, A and B. Determine, if possible, what A

and B are if they address you in the ways described. If you can not determine what these two

people are, can you draw any conclusions?

a) A says "The two of us are both knights." and B says "A is a knave."

b) A says "I am a knave or B is a knight" and B says nothing.

3. Five friends have access to a chat room. Is it possible to determine who is chatting if the

following information is known? Either Kevin or Heather, or both, are chatting. Either Randy or

Vijay, but not both, are chatting. If Abby is chatting, so is Randy. Vijay and Kevin are either both

chatting or neither is. If Heather is chatting, then so are Abby and Kevin. Explain your reasoning.

4. Find a compound proposition involving the propositions p, q, and r that is true when

exactly two of p,q,and r are true and is false otherwise.

Logic2 & Number Theory 1

1. Translate these statements into English, where R(x)is “x is a rabbit” and H(x)is “x hops”

and the domain consists of all animals.

a) ∀x(R(x)→H(x)) b) ∀x(R(x)∧H(x)) c) ∃x(R(x)→H(x)) d) ∃x(R(x)∧H(x))

2. Let P(x), Q(x), R(x), and S(x)be the statements “x is a duck,” “x is one of my poultry,”

“x is an officer,” and “x is willing to waltz,” respectively. Express each of these statements

using quantifiers; logical connectives; and P(x),Q(x),R(x), and S(x).