Question

How do I show that a set inequality in R2 containing both x and y is open?

Ex: 3 < x^2 + y^2 < 9

Answer #1

1. Consider the set U={(x,y) in R2| -1<x<1 and y=0}. Is U
open in R2? Is it open in R1? Is it open as a subspace of the disk
D={(x,y) in R2 | x^2+y^2<1} ?
2. Is there any subset of the plane in which a single point set
is open in the subspace topology?

Prove that the open rectangle in R2
S = { (x,y) | 2 < x<5 -8 < y <
-1}
is an open set in R2, with
the usual Euclidean distance metric.

1. Let f : R2 → R2, f(x,y) = ?g(x,y),h(x,y)? where g,h : R2 → R.
Show that
f is continuous at p0 ⇐⇒ both g,h are continuous at p0

6) Solve the inequality. Write the answer using both set-builder
notation and interval notation. Graph the solution set on a number
line. Multiply both sides of the inequality by the LCD first to
clear fractions. -3/4 x≤9

Exercise 9.1.11 Consider the set of all vectors in R2,(x, y)
such that x + y ≥ 0. Let the vector space operations be the usual
ones. Is this a vector space? Is it a subspace of R2?
Exercise 9.1.12 Consider the vectors in R2,(x, y) such that xy =
0. Is this a subspace of R2? Is it a vector space? The addition and
scalar multiplication are the usual operations.

Linear Algebra Problem:
Show that if x and y are vectors in R2, then x+y and
x-y are the two diagonals of the parallelogram whose sides are x
and y.
Thank you

how do i show the lim as x approaches 0 of ((x^2-9)(x-3)/|x-3|)
by the squeeze theorem

Problem 3
For two relations R1 and
R2 on a set A, we define the
composition of R2 after R1
as
R2°R1 = { (x,
z) ∈ A×A | (∃ y)( (x,
y) ∈ R1 ∧ (y, z) ∈
R2 )}
Recall that the inverse of a relation R, denoted
R -1, on a set A is defined as:
R -1 = { (x, y) ∈
A×A | (y, x) ∈ R)}
Suppose R = { (1, 1), (1, 2),...

For V = [2(x^2)y] i-( (z^3) + y) j + (3xyz) k, show that Stokes'
theorem
holds by calculating both sides of the equation for a square in the
x-y plane
with corners at (0; 0; 0), (3; 0; 0), (3; 3; 0), (0; 3; 0) .
Confirm that Stokes' theorem only depends on the boundary line by
integrating over the surface of a cube with an open bottom The
bounding line is the same as before.

How do I find the concentration of diluted unknown when the
y=0.2025x+0.001 and the r2 value is 0.9998

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