Question

**Remember that the DOMAIN is INTEGERS
(....,-2,-1,0,1,2,3,4.......) and the TARGET is POSITIVE INTEGERS
(1,2,3,4,.....) .**

Give explanation and proofing for each .

Find a function whose domain is the **set of all
integers** and whose **target is the set of all
positive integers** that satisfies each set of
properties.

(a) Neither one-to-one, nor onto.

(b) One-to-one, but not onto.

(c) Onto, but not one-to-one.

(d) One-to-one and onto.

Answer #1

Let A = {1, 2, 3, 4, 5, 6}. In each of the following, give an
example of a function f: A -> A with the indicated properties,
or explain why no such function exists.
(a) f is bijective, but is not the identity function f(x) =
x.
(b) f is neither one-to-one nor onto.
(c) f is one-to-one, but not onto.
(d) f is onto, but not one-to-one.

For each problem below, either give an example of a function
satisfying the give conditions, or explain why no such function
exists.
(a) An injective function f:{1,2,3,4,5}→{1,2,3,4}
(b) A surjective function f:{1,2,3,4,5}→{1,2,3,4}
(c) A bijection f:N→E, where E is the set of all positive even
integers
(d) A function f:N→E that is surjective but not injective
(e) A function f:N→E that is injective but not surjective

Consider the set {1,2,3,4}.
a) make a list of all samples of size 2 that can be drawn from
this set of integers( Sample with replacement; that is, the first
number is drawn, observed, and then replaced [returned to the
sample set] before the next drawing)
b) construct the sampling distribution of sample means for
samples of size 2 selected from this set. Provide the distribution
both in the form of a table and histogram.
c) Find μX and
σX

In each part below, sketch a graph of a function whose domain is
[0, 4] that has the desired property. No justification is needed in
any part.
(a) f(x) has an absolute maximum and absolute minimum on [0,
4].
(b) g(x) has neither an absolute maximum nor an absolute minimum
on [0, 4].
(c) h(x) has exactly two local minima and one local maximum on
[0, 4].
(d) k(x) has one inflection point and no local extrema on [0,
4].

f(x)=5x^(2/3)-2x^(5/3)
a. Give the domain of f
b. Find the critical numbers of f
c. Create a number line to determine the intervals on which f is
increasing and decreasing.
d. Use the First Derivative Test to determine whether each
critical point corresponds to a relative maximum, minimum, or
neither.

1. Let A = {1,2,3,4} and let F be the set of all functions f
from A to A. Prove or disprove each of the following
statements.
(a)For all functions f, g, h∈F, if f◦g=f◦h then g=h.
(b)For all functions f, g, h∈F, iff◦g=f◦h and f is one-to-one
then g=h.
(c) For all functions f, g, h ∈ F , if g ◦ f = h ◦ f then g =
h.
(d) For all functions f, g, h ∈...

Discrete Mathematics
(a) Let P(x) be the predicate “−10 < x < 10” with domain
Z+ (the set of all positive integers). Find the truth set of
P(x).
(b) Rewrite the statement Everybody trusts somebody in formal
language using the quantiﬁers ∀ and ∃, the variables x and y, and a
predicate P(x,y) that you must deﬁne.
(c) Write the negation of the statement in (b) both formally and
informally.

2. There is a famous problem in computation called Subset Sum:
Given a set S of n integers S = {a1, a2, a3, · · · , an} and a
target value T, is it possible to find a subset of S that adds up
to T? Consider the following example: S = {−17, −11, 22, 59} and
the target is T = 65. (a) What are all the possible subsets I can
make with S = {−17, −11, 22,...

For each problem, say if the given statement is True or False.
Give a short justification if needed.
Let f : R + → R + be a function from the positive real numbers
to the positive real numbers, such that f(x) = x for all positive
irrational x, and f(x) = 2x for all positive rational x.
a) f is surjective (i.e. f is onto).
b) f is injective (i.e. f is one-to-one).
c) f is a bijection.

Graph: f(x)= 2
Find (a) The domain of f.
(b) All asymptotes (write the equations)
(c) X- intercept(s), Y- intercept. Write them as ordered
pairs.
(d) Graph the function. Include asymptotes

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