(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable.
(2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable.
(3) Show that the intersection of two countable sets is
countable.
(4) Show that the set of all irrational numbers is
uncountable.
(5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · ·
is...
Write a program that takes a set of m numbers up to size
n and save them in an array. The program then allows you to search
for any number in the array with O(1).
Given two functions, M(x, y) and N(x, y), suppose that (∂N/∂x −
∂M/∂y)/(M − N)
is a function of x + y. That is, let f(t) be a function such
that
f(x + y) = (∂N/∂x − ∂M/∂y)/(M − N)
Assume that you can solve the differential equation
M dx + N dy = 0
by multiplying by an integrating factor μ that makes it exact
and that it can also be
written as a function of x + y,...
Suppose we have a substring of length m and text of
size n. Write an algorithm to find out if the substring is
present in the text or not. What is the complexity of your
algorithm in terms of m and n.
How to solve using functions in TI84:
In order to set rates, an insurance company is trying to
estimate the number of sick days that full time workers at an auto
repair shop take per year. A previous study indicated that the
standard deviation was 2.8 days. How large a sample
must be selected if the company wants to be 95% confident that the
true mean differs from the sample mean by no more than 1
day?
A)141 B)31 C)1024...
Write and test a C implementation of the Mobius function M(n)
defined as follows
M(n) = 1 if n = 1
= 0 if any prime factor is contained in N more than once
= (‐1)p if n is the product of p different prime factors
Examples
M(78) = ‐1 78 = 2 * 3 *
13
M(34) = 1 34 = 2 * 17
M(45) = 0 45 = 3 * 3 *
5
The first values of M(n)...
Consider the relation R defined on the set Z as follows: ∀m, n ∈
Z, (m, n) ∈ R if and only if m + n = 2k for some integer k. For
example, (3, 11) is in R because 3 + 11 = 14 = 2(7).
(a) Is the relation reflexive? Prove or disprove.
(b) Is the relation symmetric? Prove or disprove.
(c) Is the relation transitive? Prove or disprove.
(d) Is it an equivalence relation? Explain.