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: greggo@math.cmu.edu
Class webpage: http:/www.cmu.edu/blackb
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
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
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
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
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
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
21-127 Practice Problems for Midterm 1
Page 1 of 2
21-127 : Concepts of Mathematics
Fall 2016
Practice Problems for Midterm 1
Disclaimer. The actual exam will contain 5 main problems, each worth 10 points, and
a bonus problem, worth 5 points. Some of the
21-127 Assignment 2
Page 1 of 2
21-127 : Concepts of Mathematics
Fall 2016
Assignment 2
Due: Thursday, September 15, at the beginning of your recitation.
You should submit your homework in class on the due date just before the recitation begins.
Please re
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,
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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
21-127 Concepts of Mathematics
Fall, 2014
Homework 6
Instructions: Please turn in your HW at the start of recitation on Thursday, October 23rd. Use
standard size paper, write on one side, and do the problems in the indicated order. Staple your work
togeth
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) =
Recitation 2
21-127 Concepts of Math
08.30.2012
1 = 1 ?!
1=
1=
p
(1) (1) = 1 1 = i i = i2 = 1
(Note: We wont have cause to discuss the complex numbers C in this course, just in this example.)
What went wrong? Context is very important. We assume (as one o
Recitation 7
21-127 Concepts of Math
09.18.2012
Sets Proofs and Double-Containment Arguments
Suppose that S is a set and A P(S). Prove that
[
XS
XA
Let A and B be sets, in the context of a universal set A, B U . Consider the following set inequality
A B