Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld
October 7
revised October 9, 2005, 719 minutes
Solutions to In-Class Problems Week 5, Fri.
Problem 1. Figures
6.042/18.062J Mathematics for Computer Science
Tom Leighton and Marten van Dijk
December 17, 2008
Final Exam
Problem 1. [25 points] The Final Breakdown
Suppose the 6.042 nal consists of:
36 true/false questions worth 1 point each.
1 induction problem wo
inal
2
Problem 1. [8 points] Prove that for all n 2 N, the following identity holds
n
X
i2 =
i=1
Solution. By induction on n
n(n + 1)(2n + 1)
.
6
1, with induction hypothesis
P (n) :
n
X
i=0
i2 =
n(n + 1)(2n + 1)
6
for all n 2 N
ase case (n = 1):
Inductiv
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 9
revised September 8, 2005, 1167 minutes
Solutions to In- lass Problems Week 1, Fri.
Problem 1. lber
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 21
revised October 21, 2005, 1019 minutes
Solutions to In- lass Problems Week 7, Fri.
Problem 1. By now
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 3
revised October 4, 2005, 667 minutes
Solutions to In- lass Problems Week 5, Mon.
Problem 1. There is
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 16
revised September 20, 2005, 1273 minutes
Solutions to In- lass Problems Week 2, Fri.
Problem 1. Su
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 5
revised October 5, 2005, 1119 minutes
Solutions to In-Class Problems Week 5, Wed.
Problem 1. Let Ln b
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 28
revised September 29, 2005, 934 minutes
Solutions to In-Class Problems Week 4, Wed.
Problem 1. Rec
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 12
revised October 11, 2005, 598 minutes
Solutions to In-Class Problems Week 6, Wed.
Problem 1. This pr
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 7
revised
ugust 30, 2005, 955 minutes
Solutions to In- lass Problems Week 1, Wed.
Problem 1. Identify
6.042/18.062J Mathematics for Computer Science
Tom Leighton and Marten van Dijk
October 27, 2010
Midterm
Problem 1. [10 points]
Consider these two propositions:
: (
_ B) ) C
Q: (C ) ) _ (C ) B)
Which of the following best describes the relationship betwee
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 23
revised September 19, 2005, 1433 minutes
Solutions to In-Class Problems Week 3, Fri.
Problem 1. Gi
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 24
revised October 24, 2005, 1211 minutes
Solutions to In- lass Problems Week 8, Mon.
Problem 1. Given
6.042/18.062J Mathematics for Computer Science
Tom Leighton, Marten van Dijk, and Brooke Cowan
October 21, 2010
Midterm Practice Problems
Problem 1. [10 points] In problem set 1 you showed that the nand operator by itself can be
used to write equivalent e
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 12
revised September 12, 2005, 1214 minutes
Solutions to In- lass Problems Week 2, Mon.
p
Problem 1.
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 14
revised September 13, 2005, 1279 minutes
Solutions to In- lass Problems Week 2, Wed.
Problem 1. Fo
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 30
revised March 11, 2006, 1239 minutes
Solutions to In- lass Problems Week 4, Fri.
Problem 1. (a) Fo
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 19
revised October 18, 2005, 704 minutes
Solutions to In-Class Problems Week 7, Wed.
Problem 1. Lets tr
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 14
revised October 11, 2005, 701 minutes
Solutions to In- lass Problems Week 6, Fri.
Problem 1. This pr
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
October 28
revised October 27, 2005, 988 minutes
Solutions to In- lass Problems Week 8, Fri.
Problem 1. There i
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. lbert R. Meyer and Prof. Ronitt Rubinfeld
September 26
revised September 26, 2005, 1050 minutes
Solutions to In-Class Problems Week 4, Mon.
Problem 1. In
Final xam
2
Problem 1. [13 points] Give an inductive proof that the Fibonacci numbers Fn and Fn+1
are relatively prime for all n 0. The Fibonacci numbers are dened as follows:
F0 = 0
F1 = 1
Fn = F n
1
+ Fn
2
(for n
2)
Solution. We use induction on n. Let