Solutions to Problem Set 6
Problem 1. Youve seen how the RSA encryption scheme works, but why is it hard to
break? In this problem, you will see that nding secret keys is as hard as nding the
prime factorizations of integers. Since there is a general cons
Problem Set 4
Due: October 12
Reading: Week 5 Notes and Undirected Graphs section of Week 4 Notes
Problem 1. For functions f : A B and g : B C, the composition of g and f , written
g f , is the function h : A C where
h(a) := g(f (a).
(a) Prove that if f a
Problem Set 2
Due: September 26
Reading: Course notes on induction.
Problem 1. Use induction to prove that the following inequality holds for all integers
n 1.
1 3 5 (2n + 1)
1
2 4 6 (2n + 2)
2n + 2
Problem 2. This term in 6.042, were constantly trying to
Problem Set 7
Due: November 9
Reading: Counting, Notes I. & II.13.
Problem 1. There are 20 books arranged in a row on a shelf.
(a) Describe a bijection between ways of choosing 6 of these books so that no two adja
cent books are selected and 15bit sequenc
Problem Set 8
Due: November 21
Reading: Counting Notes, II.45; Notes on Generating Functions
Problem 1. Find the coefcients of
(a) x10 in (x + (1/x)100
(b) xk in (x2 (1/x)n .
Problem 2. Suppose a generalized World Series between the Sox and the Cardinals
Problem Set 5
3
Problem 6. Take a big number, such as 37273761261. Sum the digits, where every other
one is negated:
3 + (7) + 2 + (7) + 3 + (7) + 6 + (1) + 2 + (6) + 1 = 11
As it turns out, the original number is a multiple of 11 if and only if this sum
Solutions to Problem Set 2
Problem 1. Use induction to prove that the following inequality holds for all integers
n 1.
1 3 5 (2n + 1)
1
2 4 6 (2n + 2)
2n + 2
Solution. We use induction. Let P (n) be the proposition that:
1 3 5 (2n + 1)
1
2n + 2
2 4 6 (2n
Problem Set 9
Due: December 2
Reading: Week 12 Notes. Week 13 Notes, Sections 1 and 2 (Random Variables and the
Birthday Principle).
Problem 1. Professor Plum, Mr. Green, and Miss Scarlet are all plotting to shoot Colonel
Mustard. If one of these three ha
Problem Set 1
Due: September 21
Reading: Notes for Week1 and Week 2
Problem 1. A real number r is called sensible if there exist positive integers a and b such
that a/b = r. For example, setting a = 2 and b = 1 shows that 2 is sensible. Prove that
3
2 is
Problem Set 3
Due: October 3
Reading: Week 4 Notes.
Problem 1. (a) List all the different binary relations on the set cfw_0, 1.
(b) Over the domain cfw_0, 1, which of these relations are weak partial orders? strict partial
orders? equivalence relations?
P
Solutions to Problem Set 3
Problem 1. (a) List all the different binary relations on the set cfw_0, 1.
Solution. There are altogether 16 binary relations.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
cfw_(0, 0)
cfw_(0, 1)
cfw_(1, 0)
cfw_(1, 1)
c
Problem Set 6
Due: October 26
Reading: Notes 7 on State Machines & Notes 8 on Series and Asymptotics.
Problem 1. Youve seen how the RSA encryption scheme works, but why is it hard to
break? In this problem, you will see that nding secret keys is as hard a
Solutions to Problem Set 7
Problem 1. There are 20 books arranged in a row on a shelf.
(a) Describe a bijection between ways of choosing 6 of these books so that no two adja
cent books are selected and 15bit sequences with exactly 6 ones.
Solution. There
Problem Set 10
Due: December 9
Reading: Lecture notes for weeks 13 and 14.
Problem 1. MIT students sometimes delay laundry for a few days. Assume all random
values described below are mutually independent.
(a) A busy student must complete 3 problem sets b
Solutions to Problem Set 1
Problem 1. A real number r is called sensible if there exist positive integers a and b such
that a/b = r. For example, setting a = 2 and b = 1 shows that 2 is sensible. Prove that
3
2 is not sensible. (Consider only positive rea
Solutions to Problem Set 8
Problem 1. Find the coefcients of
(a) x10 in (x + (1/x)100
Solution. x55 (1/x)45 = x10 so the coefcient is
100
55
(b) xk in (x2 (1/x)n .
Solution. xk must equal (x2 )j (1/x)( n j) for some j where 0 j n, in which case
j = (n +
Solutions to Problem Set 5
3
Problem 3. Suppose that a b (mod n) and n > 0. Prove or disprove the following
assertions:
(a) ac bc (mod n) where c 0
Solution. The proof is by induction on c with the hypothesis that ac bc (mod n). If
c = 0, then the claim h
Solutions to Problem Set 4
Problem 1. For functions f : A B and g : B C, the composition of g and f , written
g f , is the function h : A C where
h(a) := g(f (a).
(a) Prove that if f and g are bijections, then so is g f .
(b) Prove that if f : A B is a bi
Solutions to Problem Set 10
Problem 1. MIT students sometimes delay laundry for a few days. Assume all random
values described below are mutually independent.
(a) A busy student must complete 3 problem sets before doing laundry. Each problem
set requires
Solutions to Problem Set 9
Problem 1. Professor Plum, Mr. Green, and Miss Scarlet are all plotting to shoot Colonel
Mustard. If one of these three has both an opportunity and the revolver, then that person
shoots Colonel Mustard. Otherwise, Colonel Mustar
DUEL 6-WEEK PARTNER-BASED MUSCLE-BUILDING TRAINER
PREFERRED FOOD LIST
PROTEIN
o Eggs
o Canned tuna
o Deli turkey
o Salmon
o Chicken
o Beef jerky
o Steak
o Pork tenderloin
o Turkey meatballs
o Turkey bacon
o Tilapia
o Greek yogurt
o Ground turkey
o Ground