Question

In: Advanced Math

*(4) (a) Prove that if p=(x,y) is in the set where y<x and if r=distance from...

*(4) (a) Prove that if p=(x,y) is in the set where y<x and if r=distance from p to the line y=x then the ball about p of radius r does not intersect with the line y=x.

(b) Prove that the set where y<c is an open set.

Justify your answer

Solutions

Expert Solution

1A. Suppose de notes the distance between the points and .

r = distance from the point p to the line y=x .

There exist such that .

Let be any point on the line .

Now ,

The point does not lie inside the ball centered at p and radius r .

As be arbitrary point on the line . Hence the line does not intersect the ball centered at p and radius r .

1B. Let .

We will prove that the set B is open .

Let , then .

If we choose

Then the ball centered at and radius entirely contained in .

(x,y) is an interior point of B .

As (x,y ) is arbitrary so every point of B is an interior point of B .

Hence the set B is open .

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is...

Let R[x, y] be the set of polynomials in two coefficients. Prove that R[x, y] is a vector space over R. A polynomial f(x, y) is called degree d homogenous polynomial if the combined degree in x and y of each term is d. Let Vd be the set of degree d homogenous polynomials from R[x, y]. Is Vd a subspace of R[x, y]? Prove your answer.

a. Prove that y=sin(x) is a subspace of R^2 b. Prove that a set of 2x2...

a. Prove that y=sin(x) is a subspace of R^2 b. Prove that a set of 2x2 non invertible matrices a subspace of all 2x2 matrices

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there...

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there exists a point x_0 ∈ S which is “closest” to p. That is, prove that there exists x0 ∈ S such that |x_0 − p| is minimal.

Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q...

Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q such that x < q < y. Strategy for solving the problem Show that there exists an n ∈ N+ such that 0 < 1/n < y - x. Letting A = {k : Z | k < ny}, where Z denotes the set of all integers, show that A is a non-empty subset of R with an upper bound in R. (Hint: Use...

Consider the region R, which is bounded by the curves y=3x and x=y(4−y). (a) Set up, but...

Consider the region R, which is bounded by the curves y=3x and x=y(4−y). (a) Set up, but DO NOT SOLVE, an integral to find the area of the region RR. (b) Set up, but DO NOT SOLVE, an integral to find the volume of the solid resulting from revolving the region RRaround the xx-axis. (c) Set up, but DO NOT SOLVE, an integral to find the volume of the solid resulting from revolving the region RRaround the line x=−5x=−5.

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that...

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that X / R is a partition of X.

using / for integral Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is the...

using / for integral Evaluate the double integral //R cos( (y-x)/(y+x) )dA where R is the trapezoidal region with vertices (1,0), (2,0), (0,2), and (0,1)

Maximize p = x subject to x − y ≤ 4 −x + 3y ≤ 4...

Maximize p = x subject to x − y ≤ 4 −x + 3y ≤ 4 x ≥ 0, y ≥ 0. HINT [See Examples 1 and 2.] p = (x, y) = ____________

Determine whether each statement is true or false, and prove or disprove, as appropriate. (a) (∀x∈R)(∃y∈R)[xy=1].(∀x∈R)(∃y∈R)[xy=1]....

Determine whether each statement is true or false, and prove or disprove, as appropriate. (a) (∀x∈R)(∃y∈R)[xy=1].(∀x∈R)(∃y∈R)[xy=1]. (b) (∃x∈R)(∀y∈R)[xy=1].(∃x∈R)(∀y∈R)[xy=1]. (c) (∃x∈R)(∀y∈R)[xy>0].(∃x∈R)(∀y∈R)[xy>0]. (d) (∀x∈R)(∃y∈R)[xy>0].(∀x∈R)(∃y∈R)[xy>0]. (e) (∀x∈R)(∃y∈R)(∀z∈R)[xy=xz].(∀x∈R)(∃y∈R)(∀z∈R)[xy=xz]. (f) (∃y∈R)(∀x∈R)(∃z∈R)[xy=xz].(∃y∈R)(∀x∈R)(∃z∈R)[xy=xz]. (g) (∀x∈Q)(∃y∈Z)[xy∈Z].(∀x∈Q)(∃y∈Z)[xy∈Z]. (h) (∃x∈Z+)(∀y∈Z+)[y≤x].(∃x∈Z+)(∀y∈Z+)[y≤x]. (i) (∀y∈Z+)(∃x∈Z+)[y≤x].(∀y∈Z+)(∃x∈Z+)[y≤x]. (j) (∀x,y∈Z)[x<y⇒(∃z∈Z)[x<z<y]].(∀x,y∈Z)[x<y⇒(∃z∈Z)[x<z<y]]. (k) (∀x,y∈Q)[x<y⇒(∃z∈Q)[x<z<y]].(∀x,y∈Q)[x<y⇒(∃z∈Q)[x<z<y]].

Subjects