Logic 1
1. Let p, q, and r be the propositions:
p: You get an A on the final exam;
q: You do every exerci in this book;
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r: You get an A in this class.
write the propositions using q,q and r and logical connectives.
a) you get an A in this class, but you don't do every exerci in this book.
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b) you get an A on the final, you do every exerci 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 exerci in this book; nevertheless, you
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get an A in this class.
学习一对一e) Getting an A on the final and doing every exerci 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 exerci in this book or you get an A on the final.
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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 the 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 th
e 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 fal otherwi.
Logic2 & Number Theory 1
1. Translate the 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))琥珀留学 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 the statements using quantifiers; logical connectives; and P(x),Q(x),R(x), and S(x).
a) No ducks are willing to waltz.
b) No officers ever decline to waltz.
c) All my poultry are ducks.
piccolo>职称英语新增文章d) My poultry are not officers.
e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?
3. Let F(x, y)be the statement “x can fool y,” where the domain consists of all people in the world. U quantifiers to express each of the statements.
a) Everybody can fool Fred.
b) Evelyn can fool everybody.
c) Everybody can fool somebody.
d) There is no one who can fool everybody.亲子教育培训
e) Everyone can be fooled by somebody.
f) No one can fool both Fred and Jerry.
g) Nancy can fool exactly two people.
h) There is exactly one person whom everybody can fool.
i) No one can fool himlf or herlf.
j) There is someone who can fool exactly one person besides himlf or herlf.
4. Solve each of the congruences.
a) 34x ≡ 77(mod 89)
b) 144x ≡ 4(mod 233)
5. U the construction in the proof of the Chine remainder theorem to find all solutions to the system of congruences
x≡2(mod 3),x≡1(mod 4), and x≡3(mod 5).
6. Find all solutions, if any, to the system of congruences x≡5(mod 6), x≡3(mod 10), and x≡8(mod 15).
Number Theory 2
1. a)U Fermat’s little theorem to compute 3302 mod 5, 3302 mod 7, and 3302 mod 11.
b)U your results from part (a) and the Chine remainder theorem to find 3302 mod 385. (Note that 385=5·7·11.)
2. U Wilson's Theorem to prove Fermat’s little theorem.
3. Show that we can easily factor n when we know that n is the product of two primes,p and q, and we know the value of (p−1)(q−1).
4. Let n be a positive integer and let n−1=2st, where s is a nonnegative integer and t is an odd positive integer. We say that n pass Miller’s test for the ba b if either bt ≡1 (mod n) or b(2^j)t ≡−1 (mod n) for some j with 0≤j≤s−1. It can be shown (e [Ro10]) that
a composite integer n pass Miller’s test for fewer than n/4 bas b with 1<b<n. A composite positive integer n that pass Miller’s test to the ba b is called a strong pudoprime to the ba b.
