Question

1. Let A ⊆ R and p ∈ R. We say that A is bounded away from p if there is some c ∈ R+ such that |x − p| ≥ c for all x ∈ A. Prove that A is bounded away from p if and only if p not equal to A and the set n { 1 / |x−p| : x ∈ A} is bounded.

2. (a) Let n ∈ natural number(N) , and suppose that k 2 < n < (k + 1)2 for some k ∈ N. Prove that n does not have a square root in N.

(b) Let c ∈ R \ {0}. Prove that if c has a square root in Z, then c has a square root in N.

(c) Prove that 2 does not have a square root in Z

Answer #1

Let A = R. For θ1, θ2 ∈ R, we will say that θ1 ∼ θ2 if θ1 − θ2 =
2lπ for
some l ∈ Z.
By making a natural choice of equivalence class representative,
we showed that one can identify Zm with the set {0,1,··· ,m−1} .
Find A/∼, and identify it with a subset of R in a similar way.
This is a discrete math question.
Thank you for any piece of help!

A. Let p and r be
real numbers, with p < r. Using the axioms of
the real number system, prove there exists a real number q
so that p < q < r.
B. Let f: R→R be a polynomial
function of even degree and let A={f(x)|x
∈R} be the range of f. Define f
such that it has at least two terms.
1. Using the properties and
definitions of the real number system, and in particular the
definition...

Exercise 6.4. We say that a number x divides another number z if
there exists an integer k such that xk = z. Prove the following
statement. For all natural numbers n, the polynomial x − y divides
the polynomial x n − y n.

1. Let R be the rectangle in the xy-plane bounded by the lines x
= 1, x = 4, y = −1, and y = 2. Evaluate Z Z R sin(πx + πy) dA.
2. Let T be the triangle with vertices (0, 0), (0, 2), and (1,
0). Evaluate the integral Z Z T xy^2 dA
ZZ means double integral. All x's are variables. Thank you!.

Let
X be finite set . Let R be the relation on P(X). A,B∈P(X) A R B Iff
|A|＝|B| prove R is an equivalence relation

1. (a) Let S be a nonempty set of real numbers that is bounded
above. Prove that if u and v are both least upper bounds of S, then
u = v.
(b) Let a > 0 be a real number. Deﬁne S := {1 − a n : n ∈ N}.
Prove that if epsilon > 0, then there is an element x ∈ S such
that x > 1−epsilon.

(a) Let f(z) = z^2. R is bounded by y = x, y= -x and x = 1. Find
the image of R under the mapping f.
(b)Find the all values of (-i)^(i)
(c)Find the all values of (1-i)^(4i)

1. [10 marks] We begin with some mathematics regarding
uncountability. Let N = {0, 1, 2, 3, . . .} denote the set of
natural numbers.
(a) [5 marks] Prove that the set of binary numbers has the same
size as N by giving a bijection between the binary numbers and
N.
(b) [5 marks] Let B denote the set of all infinite sequences
over the English alphabet. Show that B is uncountable using a proof
by diagonalization.

Let (X, d) be a metric space, and let U denote the set of all
uniformly continuous functions from X into R. (a) If f,g ∈ U and we
define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X,
show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U
and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X,...

Let R be the region bounded above by f(x) = 3 times the (sqr
root of x) and the x-axis between x = 4 and x = 16. Approximate the
area of R using a midpoint Riemann sum with n = 6 subintervals.
Sketch a graph of R and illustrate how you are approximating it’
area with rectangles. Round your answer to three decimal
places.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago

asked 4 hours ago

asked 4 hours ago