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 bas
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
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
SI Concepts of Math
Exam 3 Review Solutions
Natalie Morris [email protected]
14 November 2010
1. Without using your notes or the book, write out the counting in two ways arguments for
Pascals and Chairp
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
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 no
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 solutio
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=
SI Concepts of Math
Final Exam Review Solutions
Natalie Morris [email protected]
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 n
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
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 assump
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:
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 b
SI Concepts of Math
Natalie Morris [email protected]
26 September 2010
Given that P Q, write the statements for
(a) Contrapositive:
(b) Converse:
1. When to use Proof of Contrapositive:
Reverse the pro
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
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
SI Concepts of Math
Natalie Morris [email protected]
26 September 2010
Given that P Q, write the statements for
(a) Contrapositive:
(b) Converse:
1. When to use Proof of Contrapositive:
Reverse the pro
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