21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, April 29
Last time:
Denition. A nite probability space is a set S together with a function P dened
on the subsets of S such that
a. For A S, 0 P (A) 1. (So P : 2S [0, 1].)
b. P (S) = 1, a
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, January 30
Next, we look at how we phrase and combine multiple simple statements to form
more interesting complicated statements. Without the associated logical connectives,
most resul
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, February 20
The following material on natural number representation was the subject of recitation
on Tuesday, February 19th - your TAs presentation likely varied from whats given here.
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, February 6
The following denitions and result were covered in recitation on Tuesday, 2/5, by your
TA. Your TAs presentation may have diered from whats here.
Denition. A sequence is a f
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, January 23
Sets, cont.
Denition (1.17): A list with entries in A consists of elements of A in a specied order
(with repetition allowed). A k-tuple is a list with k entries, and we writ
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, January 16
We begin with the Quadratic Formula. In case you werent sure, this is something
you should already know.
Ancient Babylonians Problem: given s and p, nd numbers x and y so t
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, April 8
Last time:
Denition. A relation on S is a subset of S S.
This time
One of the most important relationships between two objects in math is equality.
Equality, though, can be a litt
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, April 1
Divisibility - Factors and Factorization
We recall a denition brought up earlier in the class:
Denition. If a, b Z with b = 0, and a = mb for some integer m, then b divides a,
wri
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, April 15
The Principle of Inclusion-Exclusion
Example (Eulers totient ). For a positive integer m, let (m) be the number of
integers in [m] which are relatively prime to m. This function
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, March 18
Last time:
A set A is countable if a bijection f : N A exists, i.e. sequence containing each
element exactly once.
If A is innite and a surjection g : N A exists, then A is cou
21-127 Concepts of Mathematics, Spring 2013
Homework 1 Solutions
1.22 - From the solutions manual:
1.27 - From the solutions manual:
1.29 - From the solutions manual:
1.33 - From the solutions manual:
1.34 - From the solutions manual:
1.36 - From the solu
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, February 27
From last time:
Denition. If A, B are subsets of R and f : A B, we say that f is
strictly increasing if f (x) < f (y) for all x, y A with x < y.
strictly decreasing if f
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, April 17
Last time:
Theorem (The Principle of Inclusion-Exclusion). Given a nite universe U, and subsets
A1 , . . . , An , the number N of elements of the universe not in any set Ai is
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Friday, April 26
Last time:
Theorem (The Pigeonhole Principle). If we place more than kn objects into n classes,
then some class has more than k objects.
This time:
The Pigeonhole Principle often
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, April 15
The Principle of Inclusion-Exclusion
Example (Eulers totient ). For a positive integer m, let (m) be the number of
integers in [m] which are relatively prime to m. This function
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Friday, May 2
Last time:
Denition. If A and B are events in a probability space and P (B) = 0, then the
conditional probability of A given B is
P (A|B) =
P (A B)
.
P (B)
Problem. Suppose we have
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, April 10
Last time:
Congruence mod n is an equivalence relation, and its set of equivalence classes
(congruence classes) is denoted Zn .
|Zn | = n - each element of Zn is a set of th
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, March 20
Counting Finite Sets for Fun and No Prot
After having spent several lectures discussing sizes of innite sets, its time to switch
back to nite sets and actually count some thin
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, March 6
Denition. We say two sets A and B have the same cardinality if there exists a
bijection from A to B. We say an innite set A is countable (or countably innite)
if there exists a
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, April 3
The following denition and result were presented in recitation on Tuesday, 4/2.
Denition (Denition 6.11). Given integer a, b, not both 0, the greatest common
divisor gcd(a, b)
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Wednesday, March 27
Last time:
There are n(n 1) (n k + 1) arrangements of k distinct items from an nelement set. (An arrangement is just a list.)
A selection of k elements from [n] is a k-eleme
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, March 4
Last time:
Recall that for k N, [k] = cfw_1, 2, . . . , k, and we dene [0] = . We say a set A is
nite if there exists k N cfw_0 and a bijection f : A [k].
Proposition (Prop. 4.37)
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, January 28
Quantiers and Logical Statements
The proofs covered Friday serve a number of purposes in this course, and the rst is
the emphasis on precise language; today we we begin our loo
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Monday, February 4
Induction
Our next topic is the method of induction, a more sophisticated proof technique
than those discussed on Friday, and one which you will see time and time again if you
21-127 Concepts of Mathematics, Spring 2013
Lecture Notes - Friday, January 25
The last few lectures have introduced some concepts and denitions that well see
throughout the course. We are next going to turn our attention towards formal logic and
the lang
L20: Assembly Programming 2
(Subroutines)
18-240: Structure and Design of Digital Systems
Don Thomas & Bill Nace
Spring 2013
2004 - 2013. All Rights Reserved. All work contained herein is
copyrighted and used by permission of the authors. Contact
ece240-
L15: Register-Transfer Datapath #2
18-240: Structure and Design of Digital Systems
Bill Nace & Don Thomas
Spring 2013
2004 - 2013. All Rights Reserved. All work contained herein is
copyrighted and used by permission of the authors. Contact
ece240-staff@e
L8: Combinational Logic Wrap-up
18-240: Structure and Design of Digital Systems
Bill Nace & Don Thomas
Spring 2013
2004 - 2013. All Rights Reserved. All work contained herein is
copyrighted and used by permission of the authors. Contact
ece240-staff@ece.
L7: Number Systems and Arithmetic
18-240: Structure and Design of Digital Systems
Bill Nace & Don Thomas
Spring 2013
2004 - 2013. All Rights Reserved. All work contained herein is
copyrighted and used by permission of the authors. Contact
ece240-staff@ec
L3: Introduction to SystemVerilog
18-240: Structure and Design of Digital Systems
Don Thomas & Bill Nace
Spring 2013
2004 - 2013. All Rights Reserved. All work contained herein is
copyrighted and used by permission of the authors. Contact
ece240-staff@ec