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.
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.
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 - you
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 h
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 al
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 Baby
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 objec
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 =
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
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 i
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:
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
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 nu
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 th
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
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
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 deno
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 t
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
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
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
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 cf
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
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,
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
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
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 us
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
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 use
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 use