21-127 Assignment 2 Solutions
Page 1 of 3
1. Quantiers.
(a) If x > 0 such that g(x) f (x) then x0 > 0 we have f (x0 ) M .
(b) f (x) is a function dierentiable at any x on the interval [a, b], and there exists an
x in [a, b] where f (x) is discontinuous at
21-127 Assignment 1 Solutions
Page 1 of 3
1. Truth Tables.
p
T
T
F
F
q (p q) (p q)
T
F
F
F
T
F
F
F
2. Logical statements.
(a) First use x y x y to eliminate implication.
(p q) (p q) (p q) (p q)
Then use the de Morgans laws.
(p q) (p q)
Then use associat
21-127 Assignment 3
Page 1 of 2
21-127 : Concepts of Mathematics
Fall 2015
Assignment 3
Due: Thursday, September 24, at the beginning of your recitation.
Last Name:
First Name:
Andrew ID:
Recitation:
You should submit your homework in class on the due dat
21-127 Assignment 3 Solutions
Page 1 of 4
1. Prove that 3 2 is an irrational number.
Since ( 3 2)( 3 + 2) = 1, we conclude that 3 2 and 3 + 2 are both either
rational or
irrational. Let us assume that 3 2 is rational. If we subtract them, we
But the
get
21-127 Assignment 1
Page 1 of 3
21-127 : Concepts of Mathematics
Fall 2015
Assignment 1
Due: Thursday, September 10, at the beginning of your recitation.
Last Name:
First Name:
Andrew ID:
Recitation:
You should submit your homework in class on the due dat
21-127 Concepts of Mathematics, Spring 2014, MIDTERM
II-alternate-SOLNS
Name:
Andrew ID:
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive News
21-127: MIDTERM II Extra Practice Problems
Solutions will be covered in class on Friday
These are problems that were on our midterm list and didnt make the nal cut. They are all
at the level of problems that youll see on the midterm. For each problem, unl
21-127 Concepts of Mathematics
Spring 2014
Test 3 Makeup Solutions
Andrew ID:
Name:
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newstead
21-127 Concepts of Mathematics
Spring 2014
Test 1 Solutions
Andrew ID:
Name: Solutions
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newste
Lecture Notes, Concepts of Mathematics (21-127)
Lecture 1, Recitation AD, Spring 2008
3
3.3
Relations
Relations, Equivalence Relations, and Partitions
Recall the denition of the Cartesian product X Y of two sets X and Y . From basic set theory, we have: T
21-127 Concepts of Mathematics, Spring 2014, MIDTERM II
SOLUTIONS
Name:
Andrew ID:
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newstead
S
21-127 Concepts of Mathematics
Spring 2014
Test 3 Solutions
Andrew ID:
Name:
Please circle your section:
Professors Picollelli and Harchol-Balters Sections:
Section A
12:30
Clive Newstead
Section B
1:30
Reilly Miller
Section F
10:30
Clive Newstead
Section
21-127 Concepts of Mathematics, Spring 2014
Homework 10 Solutions
1. Proof: We prove this by strong induction on n. Well allow for two base cases: n = 1
and n = 2.
BC: If n = 1, the assertion is that |A1 | = |A1 |, which is true. If n = 2, this assertion
21-127 Concepts of Mathematics, Spring 2014
Homework 11
Due: Friday, May 2nd
The following problems are assigned but will not be collected:
Pages 655-656 : 8.9.8 (a)-(d), 8.9.9, 8.9.16, 8.9.18, 8.9.13
Let n N. Use the Binomial Theorem to prove that
k[n]
Midterm II Review
Concepts of Mathematics
March 21, 2014
1. Fibonacci Fun
Prove the fn (3/2)n2 for all n N .
f is the bonacci seqeunce so f1 = 1, f2 = 1 and fi = fi1 = fi2 where i > 2 .
2. Greatest Common Divisor
Get the gcd(1001, 150) using the Euclidean
Midterm II Review Solutions
Concepts of Mathematics
March 23, 2014
1. Fibonacci Fun
Prove the fn (3/2)n2 for all n N .
f is the bonacci seqeunce so f1 = 1, f2 = 1 and fi = fi1 + fi2 where i > 2 .
Solution:
Let P(n) be the statement: fn (3/2)n2 .
Base case
21-127 Concepts of Mathematics, Spring 2014
Homework 8 Solutions
Solutions to the Collected Problems:
1. Solution: We will use the fact that R is an equivalence relation and therefore has the
properties of reexivity, symmetry, and transitivity.
1. Since R
21-127 Concepts of Mathematics, Spring 2014
Homework 9 Solutions
1. Solution:
(a) This function is neither injective nor surjective. To see that its not injective, note that
f (1, 5) = 29 = f (6, 1). To see that its not surjective, note that for all (x, y
21-127 Concepts of Mathematics, Spring 2014
Homework 10
Due: Thursday, April 17th
The following problems are assigned but will not be collected:
Pages 553-555 : 7.8.29, 7.8.31, 7.8.34, 7.8.35, 7.8.36, 7.8.38
The ve problems given below will be collected:
21-127 Concepts of Mathematics, Spring 2014
Homework 9
Due: Wednesday, April 9th in class
The following problems are assigned but will not be collected:
Pages 548-552 : 7.8.3, 7.8.4, 7.8.7, 7.8.11, 7.8.24, 7.8.25
The ve problems given below will be colle
21-127 Concepts of Mathematics, Spring 2014
Homework 8
Due: Thursday, April 3rd
The following problems are assigned but will not be collected:
(Modular Reasoning) What is the remainder when 2100 is divided by 13?
Pages 548-551 : 7.8.1, 7.8.13, 7.8.16, 7
21-127 Concepts of Mathematics, Spring 2014
Homework 7
Due: Thursday, March 20th
The six problems given below will be collected:
DIRECTIONS: For each problem be very careful to show all your work and in particular
to cite every theorem/lemma/proposition/a
21-127 Concepts of Mathematics, Spring 2014
Homework 5 Solutions
1. Suppose xn Z for every n N, and that xn = 2xn1 + 3xn2 for all n 3.
a. Prove that if x1 and x2 are both odd, then xn is odd for all n N.
b. Prove that if x1 = x2 = 1, then for all n N,
xn
21-127 Concepts of Mathematics, Spring 2014
Homework 11 Solutions
1. Solution: The word COMBINATORICS consists of 13 letters, with the letters C,O,I
occuring twice each, and M,B,N,A,T,R,S occurring once. Each anagram of COMBINATORICS can be determined by
Practice List Solutions
March 21, 2014
1. [Divisibility by 3] Prove that a number, x, is divisible by 3 i the sum of its digits
is divisible by 3. Start by assuming that x is a number with r + 1 digits as follows:
ar ar1 ar2 . . . a1 a0 .
Solution:
This i
CONCEPTS OF MATHEMATICS, SUMMER 1 2014
ASSIGNMENT 1
Due Friday, May 23 at the beginning of class. Make sure to
include your name, Andrew ID, and the list of your collaborators (if
any) with your assignment. You may discuss problems with others, but
you ma
2
ASSIGNMENT 1
(3) Tell me about your math education. For example, what
courses in mathematics or science have you taken before?
(4) Why are you taking 21-127? What do you expect to get out of
it?
Problem 1 (20 points)
Recall that is dened to be the ratio
ASSIGNMENT 1
3
from the lecture notes, prove that 0 6= 1. Hint: This should be
very easy.
(4) Show that not all nonzero real numbers are positive by giving
an explicit example of a negative real number. Of course, you
should prove (using only the axioms a
21-127 Assignment 6 Solutions
Page 1 of 4
1. You can interpret x1 + x2 + . . . + xk n in the following way: at most n identical
gold coins are distributed among k pirates. What if we always start with n coins, but
not necessarily distribute all of them am
21-127 Assignment 8
Page 1 of 2
21-127 : Concepts of Mathematics
Fall 2015
Assignment 8
Due: Thursday, November 12, at the beginning of your recitation.
Last Name:
First Name:
Andrew ID:
Recitation:
You should submit your homework in class on the due date