The Sheffer Stroke is introduced in exercise 10 on page 76 of
the supplementary exercises for section 3.3.
(a) What might be significant about the Sheffer Stroke for the
designer of a computer or other digital electronic hardware?
The significant about th
Functions (Home Work)
1. Define the following functions on the integers by f(k) = k + 1, g(k) = 2k, and h(k) = k/2
(a) Which of these functions are one-to-one?
(b) Which of these functions are onto?
(c) Express in simplest terms the compositions f g, g f
(a). We use mathematical notations for having a better idea of
algorithm efficiencies rather than measuring them in hard
time in order to study the properties of a particular algorithm
independently of the execution environments as they vary
greatly with
Recursion and Recurrence Relations
(Home Work)
1. Write down all derangements of the set cfw_,b,c,d and show that the number of
derangements is the same as predicted by the recurrence D(n) = (n - 1)(D(n - 2) + D(n - 1)
with initial values D(1) = 0 and D(2
1. Explain in words why the equation
true.
is
From the Right hand side of equation 1, use the
equation 2. That is substitute k with n-k and
simplify. You will get the same thing.
2. Use the definition
to show that the
equation in question (1) is true.
(nk
Recursion and Recurrence Relations (Home Work)
1. Let k = 3k + k - 2 for all k 0.
a. Write down the values of 1, 2 and 3.
b. Write down the values of A(1), A(2) and A(3) defined by the recurrence relation: A(0) = -1, A(k) = 3A(k-1) - 2k +
7, k 1.
c. Show
In linear algebra, however, you instead talk about linear transformations, which are not (I
cannot emphasize this enough) a list of numbers, although sometimes it is convenient to
use a particular matrix to write down a linear transformation. The differen
Assignment Unit 3
1. Construct the truth table of
p
q
r
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
F
F
T
T
F
F
T
T
F
T
F
T
F
T
F
T
F
F
T
T
F
F
F
F
F
F
F
T
F
F
F
T
T
F
T
F
F
F
F
F
T
F
T
T
F
F
F
T
2. Use truth tables to prove that
p
q
r
T
T
T
T
F
F
F
F
What does an identity matrix allow you to do?
An identity matrix is a square matrix of size n x n, where the diagonal elements are all 1s and the
other elements are all 0s.
An identity matrix serves as the unit of the ring of all n x n matrices and also a
Learning Journal
George Boole (1815-1864) mathematical system became known as
Boolean algebra, a world in which all other possibilities are invalid by
fiat.
Boolean algebra, is the branch of algebra in which the values of the
variables are the truth value
Relations and Graphs
1. Let A= cfw_-2, -1, 0, 1, 2. Let r be the relation defined by xry if and only if y=-x . Let S be the
relation defined by xsy if and only if y= |x|.
(a) Write down as a subset of A X A .
(b) Write down as a subset of AXA.
(c) With ro
References
The binary system is a numerical system that uses only two symbols, 0 and 1. Due
to its ease of implementation in digital electronic circuitry using logic gates, all
modern computers use the binary system internally.
The following are some typi
Learning Journal Unit 1
1. Write a one paragraph summary of what you learned about Set Theory in this unit.
In this unit 1, I earn the foundational building block of Set Theory, which are the set language,
notation Venn diagrams which are all parts of set
Book of Proof
Richard Hammack
Virginia Commonwealth University
Richard Hammack (publisher)
Department of Mathematics & Applied Mathematics
P.O. Box 842014
Virginia Commonwealth University
Richmond, Virginia, 23284
Book of Proof
Edition 2.2
2013 by Richar
Continuity and First Principles of Derivatives
Show whether the following functions
are continuous at the given value x = a,
using Cauchy:
Find the derivative using
1.
11. () = 5 2 2 + 4
2.
Mr. King
12.
3.
13.
Show, using Cauchy, how the following
piecewi
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prove from first principles that f is continuous at x = -3
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How To Construct a Delta-Epsilon Proof
The proof, using delta and epsilon, that a function has a limit will mirror the definition of the
limit. Therefore, we first recall the definition:
limxcf(x)=Llimxcf(x)=L means
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Applied Discrete Structures
Alan Doerr and Kenneth Levasseur
Department Of Mathematical Sciences
University of Massachusetts Lowell
Version 2.0
March 2013
Home
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Errata
Home: http:/faculty.uml.edu/klevasseur/ADS2/
Blog: http:/applieddiscretestructures.
1. Explain in words why the equation is true.
Answer: It is true because 'n' choose 'k', that is 'n' object to choose from, pick 'k' of those objects.
'n' objects taken 'k' at a time.
2. Use the definition to show that the equation in question (1) is true