Full Name: Section:
MATH 127: Exam 1A Wednesday, February 11, 2015
- You are expected to justify your answers in a manner that an average 21—127 student could
understand. This may require sentences for clariﬁcation.
0 Your work should be neat and organi
Full Name: K 1: Section:
MATH 127: Exam 1 Wednesday, September 25, 2013
0 You are expected to justify your answers in a manner that an average 21127 student could
understand. This may require sentences for clarication.
0 Your work should be neat an
Math 21-127
Concepts of Mathematics
Syllabus, Spring 2013
Time and Place: Wean Hall 7500
MWF 1:30-2:20, 2:30-3:20
Instructor: Dr. Greggo M. Johnson
Oce: Wean Hall 8122
Phone: 412-268-1504
E-mail: [email protected]
Class webpage: http:/www.cmu.edu/blackb
Lecture 5
21-127 Concepts of Math
09.07.2012
Administrivia
Homework 1 will be returned on Tuesday, in recitation.
Homework 2 has been posted, and is due next Thursday, in recitation.
Induction Wrap-up
Lets summarize some of our work on inductive arguments
Lecture 19
21-127 Concepts of Math
10.12.2012
Modular Arithmetic
Chinese Remainder Theorem
The Chinese Remainder Theorem: General Sun Tzu attempts to divide his troops into certain formations. He tries to make two rows of soldiers, but finds there is one
Lecture 24
21-127 Concepts of Math
10.29.2012
Finite Cardinality Result: Using Bijections
Recall these two definitions:
|A| = |B| there exists a bijection f : A B.
A is finite n N cfw_0 such that there exists a bijection f : A [n]. (On homework, you will
Recitation 22
21-127 Concepts of Math
11.13.2012
Counting Examples
(1) Find the number of ordered arrangements of 5 distinct digits from cfw_0, 1, 2, . . . , 9. Then, find the number
of such arrangements that do not place 5 and 6 adjacent to each other. C
Lecture 35
21-127 Concepts of Math
11.28.2012
Administrivia
Exam is on Friday. I will be in my office from 2:30-5:00 today, and again from 12-2 and 4-6 tomorrow. I will
also give some tips and suggestions at the end of class today.
Homework 10 will be pos
Recitation 24
21-127 Concepts of Math
11.27.2012
Recall the following formula for the number of ways to select k objects, unordered and with repetitions
allowed, from n types of objects, and its corresponding proof:
To choose k objects from n types of obj
Recitation 11
21-127 Concepts of Math
10.02.2012
Exam Handout
Regrade policy:
If you actually have a grading complaint (I think my answer deserves more points, I think you
didnt understand what I meant in this part, you misread my solution, etc.) you must
Recitation 20
21-127 Concepts of Math
11.06.2012
Cardinality and Countable Intersections
Can you identify a countably infinite sequence of sets S1 , S2 , S3 , . . . such that every set Si is infinite, and
such that S1 S2 S3 (all strict inequalities), and
Recitation 16
21-127 Concepts of Math
10.18.2012
Functions, Compositions, Bijections, Inverses
Recall the following definitions and theorem from Wednesdays lecture:
Definition: A function that is both injective and surjective is said to be bijective (or a
21-127 Assignment 5
Page 1 of 2
21-127 : Concepts of Mathematics
Fall 2016
Assignment 5
Due: Thursday, October 13, at the beginning of your recitation.
You should submit your homework in class on the due date just before the recitation begins.
Please reme
Recitation 23
21-127 Concepts of Math
11.15.2012
Counting In Two Ways
(1)
n2
n1
=
n
X
n
k=1
k
k
One interpretation: Consider selecting a president from a pool of n people and then having each other
person being FOR or AGAINST that president. Or, consider
Recitation 21
21-127 Concepts of Math
11.08.2012
Binomial Coefficients
Recall that, in lecture on Wednesday, we derived a formula for nk , the number of ways to select k objects
from a set of n objects:
n
n!
=
k
k! (n k)!
n
In other terminology, k is th
Lecture 2
21-127 Concepts of Math
08.29.2012
Administrivia
Homework 1 is posted! Check it out! Its due next Thursday, September 6.
I posted a page about using LATEX. Check that out, too! It includes a template .tex file for Homework 1.
Office Hours: Mine
21-127 Practice Problems for Midterm 2
Page 1 of 2
21-127 : Concepts of Mathematics
Fall 2016
Practice Problems for Midterm 2
Disclaimer. The actual exam will contain 5 main problems, each worth 10 points, and
a bonus problem, worth 5 points. Some of the
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Handout for recitation 12
21-127 sections A and F
25th February 2014
TA: Clive Newstead
Computational complexity of the Euclidean algorithm
In Mondays lecture you saw the Euclidean algorithm, a step-by-step process which, given a, b Z
with 0 < |a| |b|, is
21-127 Concepts of Mathematics, Spring 2014
Notes from Lecture on Wednesday, 1/15
One of the earliest topics we will study in this course is set theory. A set is a
collection of objects, and among the most important sets we will consider are sets that
con
Greatest common divisor as a product of primes
Clive Newstead, 18th March 2014
The Fundamental Theorem of Arithmetic (FTA) is all about products of primes: it tells you that
every natural number greater than 1 has a representation as a product of primes,
21-127 Assignment 3 Solutions
Page 1 of 5
1. Since ( 3 2)( 3 + 2) = 1, we concludethat 3 2 and 3 + 2 are both either
rational
or irrational.
Let us
assume
that 3 2 is rational. If we subtract them,
we get ( 3 + 2) ( 3 2) = 2 2. If 2 2 were a
rationa
21-127 Assignment 3
Page 1 of 2
21-127 : Concepts of Mathematics
Fall 2016
Assignment 3
Due: Thursday, September 22, at the beginning of your recitation.
You should submit your homework in class on the due date just before the recitation begins.
Please re
21-127 Assignment 2 Solutions
Page 1 of 4
1. Proving set inclusion 1.
(a) x B, we consider two cases: x A or x
/ A.
If x A, then x A B, and since A B C, x C.
If x
/ A, so x Ac , then x Ac B, and since Ac B C, x C.
Q.e.d.
(b) To prove that (A B) C [A (B
21-127 Assignment 1
Page 1 of 3
21-127 : Concepts of Mathematics
Fall 2016
Assignment 1
Due: Thursday, September 8, at the beginning of your recitation.
You should submit your homework in class on the due date just before the recitation begins.
Please rem
21-127 Assignment 4 Solutions
1.
Page 1 of 3
1. Note that 1 is not in the image of f (x, y) = x+y. There are no two natural numbers
x and y such that x + y = 1. Thus f is not surjective.
2. Note that for any z N we have that f (z, 1) = z. Thus f (x, y) =