Congruence mod n
Mathematics for Computer Science
MIT 6.042J/18.062J
Def: a b (mod n)
iff n|(a - b)
Congruences:
arithmetic (mod n)
example: 30
12 (mod 9)
since
9 divides 30 - 12
Albert R Meyer,
March 31, 2010
Congruence mod n
example:
66666663
WHY?
66666
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 29
revised March 29, 2010, 739 minutes
Solutions to In-Class Problems Week 8, Mon.
Problem 1.
A number is perfect if it is equal
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 29
revised March 20, 2010, 738 minutes
In-Class Problems Week 8, Mon.
Problem 1.
A number is perfect if it is equal to the sum of
Arithmetic Assumptions
Mathematics for Computer Science
MIT 6.042J/18.062J
assume usual rules for +, , - :
Intro to
Number Theory:
Divisibility, GCDs
Albert R Meyer,
March 29, 2010
a (b+c) = ab + ac, ab = ba,
(ab)c = a (bc), a a =0,
a + 0 = a, a+1 > a, .
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 19
revised March 22, 2010, 587 minutes
Solutions to In-Class Problems Week 7, Fri.
Problem 1.
Figures 14 show different pictures
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 19
revised March 12, 2010, 1326 minutes
In-Class Problems Week 7, Fri.
Problem 1.
Figures 14 show different pictures of planar gr
Planar Graphs
Mathematics for Computer Science
MIT 6.042J/18.062J
Planar Graphs
Albert R Meyer,
March 19, 2010
Albert R Meyer,
lec 7F.1
March 19, 2010
lec 7F.2
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Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 17
revised March 15, 2010, 675 minutes
Solutions to In-Class Problems Week 7, Wed.
Problem 1.
The Elementary 18.01 Functions (F18
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 17
revised March 2, 2010, 733 minutes
In-Class Problems Week 7, Wed.
Problem 1.
The Elementary 18.01 Functions (F18s) are the set
Mathematics for Computer Science
Recursive Denitions
MIT 6.042J/18.062J
Recursive Definitions
& Structural Induction
Matched Paren Strings, M
strings in M
[]
[]
[][]
[][]
[]
M, then
M
Albert R Meyer, March 17, 2010
lec 7W.9
s=
t=
s = []
t=
s=
t =[]
s = []
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 15
revised March 15, 2010, 675 minutes
Solutions to In-Class Problems Week 7, Mon.
Problem 1.
Let G be the graph below1 . Careful
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 15
revised March 2, 2010, 734 minutes
In-Class Problems Week 7, Mon.
Problem 1.
Let G be the graph below1 . Carefully explain why
Mathematics for Computer Science
MIT 6.042J/18.062J
flights need gates, but
times overlap.
how many gates needed?
Graph Coloring
Bipartite Matching
lec 7M.1
Albert R Meyer, March 15, 2010
Airline Schedule
lec 7M.2
Albert R Meyer, March 15, 2010
Conflicts
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 12
revised March 13, 2010, 1024 minutes
Solutions to In-Class Problems Week 6, Fri.
Problem 1.
Prove that a graph is a tree iff i
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 12
revised March 13, 2010, 1029 minutes
In-Class Problems Week 6, Fri.
Problem 1.
Prove that a graph is a tree iff it has a uniqu
Connected Components
Mathematics for Computer Science
MIT 6.042J/18.062J
Every graph consists of
separate connected
pieces (subgraphs) called
Graph Connectivity
Trees
Albert R Meyer, March 12, 2010
connected components
lec 6F.1
Connected Components
13
12
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 10
revised March 8, 2010, 683 minutes
Solutions to In-Class Problems Week 6, Wed.
Problem 1.
For each of the following pairs of g
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 10
revised March 2, 2010, 734 minutes
In-Class Problems Week 6, Wed.
Problem 1.
For each of the following pairs of graphs, either
Types of Graphs
Mathematics for Computer Science
MIT 6.042J/18.062J
Simple
Graph
Simple Graphs
Degrees,
Isomorphism,
Paths
Albert R Meyer, March 10, 2010
Directed Graph
next week
this week
lec 6W.1
A simple graph:
Multi-Graph
A Simple Graph
Definition:
A
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 8
revised March 8, 2010, 683 minutes
Solutions to In-Class Problems Week 6, Mon.
Problem 1.
Four Students want separate assignmen
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 8
revised March 2, 2010, 734 minutes
In-Class Problems Week 6, Mon.
Problem 1.
Four Students want separate assignments to four VI
Mathematics for Computer Science
MIT 6.042J/18.062J
Stable
Matching
Albert R Meyer, March 8, 2010
lec 6M.1
Clip art in this lecture source unknown. All rights reserved.
This content is excluded from our Creative Commons license.
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Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 5
revised March 5, 2010, 859 minutes
Solutions to In-Class Problems Week 5, Fri.
Problem 1.
The Massachusetts Turnpike Authority
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 5
revised February 28, 2010, 1326 minutes
In-Class Problems Week 5, Fri.
Problem 1.
The Massachusetts Turnpike Authority is conce
3/5/10
Euclidean Algorithm
Mathematics for Computer Science
-for GCD(a, b)
MIT 6.042J/18.062J
1. x := a, y := b.
2. If y = 0, return x &
terminate;
3. else simultaneously:
State Machines:
Derived Variables
Albert R Meyer, March 5, 2010
(x, y) := (y, rem(x
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 3
revised March 1, 2010, 826 minutes
Solutions to In-Class Problems Week 5, Wed.
By now you are very familiar with the 6.042 icon
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 5
revised February 28, 2010, 1326 minutes
In-Class Problems Week 5, Fri.
Problem 1.
The Massachusetts Turnpike Authority is conce
3/5/10
Euclidean Algorithm
Mathematics for Computer Science
MIT 6.042J/18.062J
-for GCD(a, b)
1. x := a, y := b.
2. If y = 0, return x &
terminate;
3. else simultaneously:
State Machines:
Derived Variables
(x, y) := (y, rem(x,y)
4. Go to step 2.
Albert R
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 1
revised March 1, 2010, 826 minutes
Solutions to In-Class Problems Week 5, Mon.
Problem 1.
If a and b are distinct nodes of a di
Massachusetts Institute of Technology
6.042J/18.062J, Spring 10: Mathematics for Computer Science
Prof. Albert R. Meyer
March 3
revised February 26, 2010, 1410 minutes
In-Class Problems Week 5, Wed.
By now you are very familiar with the 6.042 icon that ap
