PRINCIPLE OF MATHEMATICAL INDUCTION
(PMI)
Let N = cfw_1, 2, 3, . . . be the natural numbers.
Let Z = cfw_. . . , 2, 1, 0, 1, 2, . . . be the integers.
Let P (n) be a statement (that is either true or false) about n.
Sometimes we denote P (n) by Pn .
2.
NAME:
Prof. Girardi
11.09.10
Exam 2
Math 300
MARK BOX
problem points
1
19
2abc
12
3abc
12
4abc
12
5
15
6
15
7
15
total
100
INSTRUCTIONS:
(1) This exam is copied 2-sided.
The mark box indicates the problems along with their points.
Check that your copy of
NAME:
Prof. Girardi
09.27.10
Exam 1
Math 300
MARK BOX
problem points
1
16
2
2
3
2
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
8
12
2
total
100
INSTRUCTIONS:
(1) The mark box indicates the problems along with their points.
Check that your copy of the exam has a
Solutions
Prof. Girardi
11.09.10
Exam 2
Math 300
MARK BOX
problem points
1
19
2abc
12
3abc
12
4abc
12
5
15
6
15
7
15
total
100
Well, each problem you have seen in class or as homework. So, as a solution key, Ill just list out
the problem source. You can g
NAME:
Prof. Girardi
Th. 12.09.10 at 5:30pm
MARK BOX
problem
points
1 (parts 1 40)
40
2
5
3
5
4ab
4
5ab
4
6ab
4
4c
10
5c
10
6c
10
7
10
total
102
Final Exam
Math 300
INSTRUCTIONS:
(1) The mark box indicates the problems along with their points.
Check that y
Math 300 - Fall 2011
Informal Summary (IS)
Prof. Girardi
0. CHAPTER 0 : Preface to the Student
0.0.1. Remark (Course Goals).
Our main goals in this course are:
to improve our ability to think and reason (useful no matter what career you choice)
to write
MARK BOX problem points 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 Total 80 % 100
Math 300X.001 Spring 99
Prof. Girardi 5/1/99
Final Exam
NAME:
INSTRUCTIONS: 1. Do 10 of the 12 problems. In the above Mark Box, put an X through the numb
MARK BOX problem points 1 10 2 10 3 10 4 10 5 10 Total 40 % 100
Math 300X.001 Spring 99
Prof. Girardi 4/6/99
Exam 2
NAME:
INSTRUCTIONS: 1. Do 4 of the 5 problems. . I am doing problem numbers: 2. Use your own paper: a. write on only one side of the page b
MARK BOX problem points 1 10 2 10 3 10 4 10 5 10 Total 40 % 100
Math 300X.001 Spring 99
Prof. Girardi 2/18/99
Exam 1
NAME:
INSTRUCTIONS: 1. Do 4 of the 5 problems. . I am doing problem numbers: 2. Use your own paper: a. write on only one side of the page
NAME:
Prof. Girardi
11.22.11
Exam 2
Math 300
MARK BOX
problem points
1
25
2
25
3
25
4
25
total
100
INSTRUCTIONS:
(1) The mark box indicates the problems along with their points.
Check that your copy of the exam has all of the problems.
(2) Use your own pa
Division Algorithm
Division Algorithm for N
Remark. Think about taking an n N and dividing it by d N. What happens?
Lets look at an example: take n = 11 and divide it by d = 5 to get
1
11
=2
5
5
11
1
=2+
5
5
or equivalently
or equivalently
11 = 5 2 + 1 .
NAME:
Prof. Girardi MARK BOX problem points 1 25 2 15 3 15 4 15 5 15 6 15 total 100 INSTRUCTIONS:
10.02.07
Exam 1
Math 300
(1) The mark box indicates the problems along with their points. Check that your copy of the exam has all of the problems. (2) When
NAME:
Prof. Girardi
10.03.06
Exam 1
Math 300
MARK BOX
problem points
1
25
2
15
3
15
4
15
5
15
6
15
total
100
INSTRUCTIONS:
(1) The mark box indicates the problems along with their points.
Check that your copy of the exam has all of the problems.
(2) When