MAD 5404 Program 1
Morgan Weiss
Problem 1 Exercises
Let Fn Cnn be the unitary matrix representing the discrete Fourier transform of length n
and so FnH Cnn is the inverse DFT of length n. For example, for n = 4
1 1 1 1
1 1
1
1
2
3
1 1 2 3
and F4H = 1 1
MAD 5403 Program 5
Morgan Weiss
Written Exercise
Consider the family of linear one-step methods defined by
yn = yn1 + h(fn + (1 )fn1 )
where 0 1.
1) Identify three well-known methods that are in this family and the associated values of .
Proof. The genera
Monday, January 25
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You have a take-home quiz due now!
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You have one more eGrade quiz before the test on Friday.
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Last time we talked about the reflexive closure r(R) and the symmetric closure s(R) of a relation.
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Today we will introduce some
Friday, January 22
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You have a take-home quiz due Monday.
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Your first test is in one week.
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Try the following exercises:
Course Notes 1.13.1, 1.14.1
Rosen (7e) 9.3 #13, 15
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Last time we discussed composition of relations, the definition of Rn, a
DISCRETE MATH II QUIZ 2 - DUE MONDAY JAN. 25, 2016
1. (4 points) Prove/disprove the following statement:
Let R be a relation on the set A. If R is transitive, then Rn Rn1 for n = 2, 3, 4, . . .
2. (4 points) Prove/disprove the following statement:
The rel
DISCRETE MATH II QUIZ 2 - DUE JAN. 25, 2016
1. (4 points) Prove/disprove the following statement:
Let R be a relation on a set A. If R is transitive, then Rn Rn1 for n = 2, 3, 4, . . .
Proof:
Use mathematical induction.
Basis step: Show Rn Rn1 for n = 2 i
Wednesday, January 20
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Your first online quiz is due this Thursday at 11:59 pm.
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From last time
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2. Prove
If R, S, T are relations on A, then !
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Typical test/quiz questions
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1. Suppose R, S are reflexive
A. RS is reflexive
D. " R is reflexive
Friday, January 15
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Today we will continue to discuss operations on relations.
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Try the following exercises:
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Course notes 1.7.1, 1.7.2, 1.7.3, 1.8.1, 1.10.1, 1.10.2, 1.10.3, 1.10.4.
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Your first online quiz is available now. It covers properties o
Friday, January 8
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Today we will talk about properties of relations, and other things from the beginning of
course notes Chapter 1.
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The related material from Rosen (7e) is in 9.1 and 9.3.
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Try these exercises
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From Rosen (7e): 9.1 #3 27 odd, 51, 53
Wednesday, Jan. 6
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Go over Syllabus
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Todays lecture covers the beginning pages of Chapter 1 from the course notes; the related
material in Rosen 7e is found in Sections 9.1 and 9.3.
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From Rosen (7e) try Exercise 1 in 9.1, Exercises 1, 3, 19, 21 in 9.3.
Wednesday, January 13
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We will continue talking about properties of relations, and other things from the beginning
of course notes Chapter 1.
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The related material from Rosen (7e) is in 9.1 and 9.3.
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Try these exercises:
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From Rosen (7e): 9.1 #3 27
Monday, January 11
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Today we will talk about properties of relations.
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The related material from Rosen (7e) is in 9.1 and 9.3.
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Try these exercises
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From Rosen (7e): 9.1 #3 27 odd, 51, 53; 9.3 #1-7 odd
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From Course Notes 1.1: Exercise 1.5.1, 1.7.1,
MAD 2104 Assignment 5: Induction
Induction
Overview: Many of the proofs you will be asked to do this semester will use the
principles of induction. This method of proof is also an important concept in many
computer science applications. An understanding o
MAD 2104 Assignment 6: Set Operations & Properties of Functions
Set Operations
Overview: We now revisit sets and induction. This time we cover operations on sets
and learn methods of proving set identities.
Read:
Online Course Notes Section 4.1 Set Opera
Solutions for Assignment 1
Section 1.1, #2. (a)How many 4-digit numbers are there. i.e., numbers from 1000
to 9999?
Solution: Number of integers from 1000 to 9999 = 9999 - 1000 + 1 = 9000.
(b) How many 5-digit numbers are there that end in 1?
Solution: Nu
MAD 2104, Section 2 - Midterm 1 Solutions
1. (8 pts.) Let p and q be primes.
(a) If p and q are distinct, then nd gcd(p2 , q 3 ) and lcm(p2 , q 3 ).
Solution: The divisors of p2 are 1, p and p2 . The divisors of q 3 are 1, q, q 2 and q 3 . Since
p2 3
p =
Solutions to Assignment 3
Section 1.7 #10.
(a) As A (x) = 1 i x A, (1) = A.
A
(b) As A (x) = 0 i x Ac , (0) = Ac .
A
Section 1.7 #12.
f 1 f (x, y) = f 1 (x + y, x y)
x + y + x y x + y (x y)
,
)
2
2
= (x, y).
=(
a+b ab
,
)
2
2
2a 2b
=( , )
2 2
= (a, b).
f
MAD 2104, Section 2 - Midterm 2 Solutions, Fall 2002
1. (16 pts.) Give a precise denition of the following:
(a) A transitive relation from a set S to itself.
Solution. R is transitive if (x, y) R and (y, z) R implies (x, z) R.
(b) The degree of a vertex v
MAD 2104, Section 2 - Midterm 3 Solutions
1. Let A =
a 0
0 b
where a and b are real numbers.
(a) (8 pts.) Compute A2 and A3 . Use this to guess a general formula for An , where n P.
a2 0
0 b2
Solution:A2 =
An =
an 0
0 bn
and A3 =
a3 0
0 b3
. So the obviou
FALL 2002 : MAD2104-02
Practice Test 3
1. Let A =
1 0
. Find a formula for An (in terms of n) where n is a positive integer.
1 1
2. Let A =
a b
c d
such that ad bc = 0 and let B =
1
adbc
d b
.
c a
(a) Compute AB and BA.
(b) Find x and y such that
1 2
1 3
MAD 2104, Section 2 - Quiz #1 Solutions
1. (4 points) Let f : N N be dened by f (n) = n3 + 2.
(a) Find f (2) and f (3).
Solution: As f (n) = 2 i n = 0, it follows that f (2) = cfw_0. Similarly, since f (n) = 3
i n = 1, it follows that f (3) = cfw_1.
(b) D
MAD2104-02
Practice Test 2
1. p is the statement I will prove this by cases, q is the statement There are more
than 500 cases, and r is the statement I can nd another way.
(a) State (r q) p in simple English.
(b) State the converse of the statement in par
Homework 7 Foundations of Computational Math 1 Fall
2014
Problem 7.1
Consider the generic Conjugate Direction algorithm for solving the minimization problem
min f (x)
xRn
where f (x) = 0.5 xT Ax xT b, b Rn , and A Rnn is symmetric positive denite.
Denote
Homework 6 Foundations of Computational Math 1 Fall
2014
Problem 6.1
6.1.a. Textbook page 241, Problem 2
6.1.b. Textbook page 241, Problem 4
6.1.c. Textbook page 241, Problem 5
Material in textbook Sections 1.7 and 5.1 is useful for these problems.
Proble
Homework 4 Foundations of Computational Math 1 Fall
2014
Problem 4.1
Let A Rnn and its inverse be partitioned
A=
A1 =
A11 A12
A21 A22
A11 A12
A21 A22
where A11 Rkk and A11 Rkk .
4.1.a. Show that if, S = A22 A21 A1 A12 , the Schur complement of A with resp
Homework 3 Foundations of Computational Math 1 Fall
2014
Problem 3.1
Recall that a unit lower triangular matrix L Rnn is a lower triangular matrix with diagonal
elements eT Lei = ii = 1. An elementary unit lower triangular column form matrix, Li , is
i
an
Homework 1 Foundations of Computational Math 1 Fall
2014
Problem 1.1
Suppose the n-bit 2s complement representation is used to encode a range of integers,
2n1 x 2n1 1.
1.1.a. If x 0 then x is represented by bit pattern obtained by complementing all
of the
Homework 2 Foundations of Computational Math 1 Fall
2014
Problem 2.1
Consider the vector space R4
2.1.a. Specify a subspace of R4 with dimension 2 by giving a basis for the subspace.
2.1.b. Show that the basis for a subspace is not unique by giving anothe