Last Recitation!
21-127 Concepts of Math
12.08.2011
Review Denitions, Theorems, Proof Techniques, Notation, etc.
Here, Ive compiled all of the sections from previous recitation sheets that contain basic denitions and
results (and added/removed some stu).
Topics for the Review on Monday November 15th, 9pm, Doherty 2210
:Cardinality:
Things to know:
Classication of cardinalities into nite, countably innite, and uncountable.
How to prove two sets have the same cardinality (show there exists a
bijection bet
21-127, Concepts of Mathematics, Review Problems
Below is a collection of problems that one may nd useful when studying for
the third exam. This review sheet is intended to be used in order to master the
material after you are comfortable with it. Do not
SI Concepts of Math
Exam 3 Review Solutions
Natalie Morris namorris@andrew
14 November 2010
1. Without using your notes or the book, write out the counting in two ways arguments for
Pascals and Chairpersons Identity. Now try the Summation Identity. Make s
CARDINALITY
Finite Sets
Notation: [n] = f1; 2; :; ng; [0] = ?:
Denition of a nite set.
Denition of an innite set.
Size of a nite set A:
How do you show that a set A is nite?
How do you show that a set A is innite?
Proposition: If there is a bijection f :
Denition 1 If a and b are integers with a 6= 0, we say that a divides b if
there is an integer c such that b = ac:
When a divides b we say that a is a factor of b and that b is a multiple of
a.
The notation a j b denotes that a divides b.
We write that a
TOPICS FOR THE FINAL EXAM
For details on each topic, read your lecture notes. Everything that has been covered in
lecture could be on the final exam. You should be able to write self-contained solutions,
without using other results. You should justify eve
CHAPTER 6: DIVISIBILITY
Proposition 6.13. If a; b; k are integers, then gcd (a; b) = gcd (a
kb; b) :
See proof in lecture notes or textbook.
The Division Algorithm: If a and b are integers, with b 6= 0, then
there exist unique integers k; r such that a =
SI Concepts of Math
Final Exam Review Solutions
Natalie Morris namorris@andrew
2 December 2010
1. Prove that if 2n 1 is prime then n is prime
Proof by contrapositive: If n is not prime, then 2n 1 is not prime.
If n is not prime, then n = ab, where a, b ar
NAME:
SECTION: SOLUTb'JS
MATH 21127, CONCEPTS 0F MATHEMATICS
Lecture 1, Fall 2010
A: Emily Allen (8:30), B, C: Brendan Sullivan (12:30, 1:30), D: Owen Traeholt (3:30)
E: Eric Ramos (4:30)
Midterm 3
Total: 100 points
a You must Show ALL work for f
21-127: Concepts of Mathematics
Homework 7 Solutions
A15 (10 points)
(a) Prove that
m3 = 6
m
m
+6
+m
3
2
for every m N by counting a set in two ways.
(b) Use part (a) to prove that
n
3
i =
i=1
n(n + 1)
2
2
for every n N.
(c) Prove the claim in part (b) by
21-127: Concepts of Mathematics
Homework 8 Solutions
6.4 (10 points)
Suppose that gcd(a, b) = 1. Prove that gcd(na, nb) = n.
Solution:
Consider the prime factorizations of a and b and n. By our assumption that gcd(a, b) = 1, we know
a and b share no commo
21-127: Concepts of Mathematics
Homework 10 Solutions
9.1 (10 points)
Let S be a probability space and let P : S R be a probability function. Let A, B S. Prove that if
A B, then P (A) P (B).
Solution:
Notice that (B A) A = , and (B A) A = B, so we can app
21-127: Concepts of Mathematics
Homework 9 Solutions
A17 (10 points)
Let a, b, c be integers such that a2 + b2 = c2 .
(a) Is it always true that at least one of cfw_a, b is even?
(b) For c divisible by 3, prove that a and b are both divisible by 3.
Soluti
SI Concepts of Math
Natalie Morris namorris@andrew
26 September 2010
Given that P Q, write the statements for
(a) Contrapositive:
(b) Converse:
1. When to use Proof of Contrapositive:
Reverse the proof - does it make sense?
Will it reduce the number of
21-127, Concepts of Mathematics, Review Problems
1. Prove the following using either weak or strong induction:
(a) 2n n + 1.
(b) F (n) =
(c)
(d)
n n
+
+
n
3
k=1 k
n1 3
k=1 k
=(
<
where F (n) = F (n 1) + F (n 2) , =
n
k=1
1 5
2
and + =
1+ 5
.
2
k)2 .
n4
.
Exam 2 Review
A: Methods of Proof
Given that P Q, write the statements for
(a) Contrapositive:
Q P
(b) Converse:
QP
When to use Proof of Contrapositive:
Reverse the proof - does it make sense?
Will it reduce the number of cases?
Keywords: not,either,bo
SI Concepts of Math
Natalie Morris namorris@andrew
26 September 2010
Given that P Q, write the statements for
(a) Contrapositive:
(b) Converse:
1. When to use Proof of Contrapositive:
Reverse the proof - does it make sense?
Will it reduce the number of
21-127: Concepts of Mathematics
Homework 6 Solutions
5.18 (10 points)
Count the number of 6 card hands, dealt from a standard deck of 52 cards, that have at least one card
in every suit.
Solution:
To have a six card hand with at least one card in every su