-[2.16] Let S be a nonempty set in R" and let x e S. Consider the set C = {y : y 2(х - х), A20, х E S}. Show that C is a cone and interpret it geometrically. b. Show that C is convex if S is convex. Suppose that S is closed. Is it necessarily true that C is closed? If not, under what conditions would C be closed? а. с.

Q: 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x,y) Sr} Show that B,(r) is closed set…

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F open?

A: A set is open if and only if it contains all of its limit points. Also recall that the union of open…

Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

A:

Q: [3.12] Let S be a nonempty convex set in R", and let f. R" → R be defined as follows:

A: Let S be a nonempty convex set in Rn, and  Let f:Rn→R be defined as follows:…

Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

A:

Q: Consider the sets S1 = {(x, y) : x^2 + y^2 < 1} and S2 = {(x, y) : y ≥ −x − 2, x ≤ 0, y ≤ 0} (a)…

A:

Q: let H = {n² |n E z*}, determine whether it is closed under

A: Given : H={n2|n∈Z+}   To Check:  H is closed under binary addition or not

Q: let H = {n² |n e Z*}, determine whether it is closed under

A: Here, we have H=n2| n∈ℤ+. To show the set is closed under binary multiplication. This means we have…

Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

A:

Q: of all Z € C that z iz e = e a Descnibe and sketch the set

A:

Q: 21. Let A and B be two non-empty bounded sets of positive numbers and let C = {xy : x e A and y e…

A:

Q: O Let X,X. X3 beasamPle of size n=3From the Bernolli Pmf Assune Ti ヒ(X、メ,メ) X、+メz+Xg and こ I= X,X t…

A: Using the Neymann factorisation theorem we can find out the sufficient statistic.

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

A:

Q: 3. Find all the fixed points and classify cach of them: f(r,u) ryev,

A: Solution:- Fx,y=xye-x2-y2 Put Fx=0 and Fy=0 Fx=e-x2-y2y-2x2=0and Fy=y-2y2 e-x2-y2=0 So,  0,0,12,12,…

Q: Blank 1 is movement of a side of a preimage parallel to that side with respect to one other fixed…

A: We have to fill the blanks.

Q: 5. If r = (r, (x, y, z) and ro = (xo, Yo, 20), describe precisely the set of all points (r, y, z)…

A: We plug the given values in the given equation and simplify

Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

A:

Q: Which of the following sets are open, and which are closed? Explain your answers by sketching the…

A:

Q: 3. Let r>0 and define the closed ball by B,(1) = {y €X : d(r,y) Sr} Show that B,(r) is elosed set…

A: Given that X be a metric space with metric d. The given topology on X is defined as:…

Q: - Show that the area of the triangle whose vertices are the [x1 yi det X2 Y2 A - points (x₁, y₁),…

A:

Q: Q: In usual topological space (R.Tu). answer the following: 1- Is [0,1] is compact set in R? Why?…

A: Note: Hi! Thank you for the question, as per the honor code, we are allowed to answer three…

Q: 10. The Proof of Theorem 6.21. Use the ideas from Exercise (9) to prove The- orem 6.21. Let A, B,…

A:

Q: C.1 Let r > 0 and define the closed ball by B,(x) = {y € X : d(x,y) <r} Show that B(x) is closed set…

A: Given: τ=B⊆X : ∀x0∈B, ∃rx0&gt;0, such that Brx0x0⊂B is a topology in X. Open ball Br(x)=y∈X : d(x,…

Q: Let W = Lin and U= Lin 2 (a) Show that R' = W U. Explain.

A: Note:- As per our guidelines, we can answer the first part of this problem as all the parts are…

Q: snip

A: Inner product of two vectors is equal to sum of product of corresponding elements of the two…

Q: The area bounded by the curve y = 7x and the lines z= 2 and =9 above the z-aris is O 539.00 square…

A:

Q: an open ball in the set X, with its radius & satisfying -d(x, a), Vx € X at the open ball Be (x)…

A:

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A:

Q: For nonempty sets A, B, and C, let f: A > B and g : B >C. Prove: If g o f is one-to-one, then fis…

A:

Q: 3) Consider the topology 7 = {G CR: 0 ¢ G or G = R} on R. Is open T 3= interval (-1,1) compact in…

A:

Q: has

A: We have to choose the correct option from the given options according to the given information

Q: (ii) Find a connected subset of R² which is not convex.

A: Note: Hi! Thank you for your question. As per the guideline we can solve one question at a time in…

Q: Show that the interval [0,1] is a closed set in R

A:

Q: For p = (1, 0) and x = (x₁,x2,...,xn) ER", define ||2|p:= - ². |P Show that n jyjx|py|q for x, y…

A:

Q: 7. Which of the following sets are open, and which are closed? Explain your answers by sketching the…

A: A set is open if all points of this set are interior points. A point x belongs to a set A  is said…

Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).

A:

Q: Sketch the following sets of points in the x-y plane. {(x, x^2): x is in R}

A:

Q: جليلة Let I = [= 1,₂1] ISR interval w that I is negligible Set.

A:

Q: (c) Given the set T := {(x, y) = R²: |xy| ≤ 1}. Is T a compact set? Show your working. If you say it…

A: Given c) T=x,y∈R2:xy≤1d) Given a metric space X,ρ and two compact subsets S,T∈X

Q: Given the set T := {(x, y) ∈ R^2 : |xy| Less or equ 1}. Is T a compact set? Show your working. If…

A:

Q: 1. For what values of k is the null space of A = k -1 a line?

A:

Q: For nonempty sets A, B, and C, let f : A –> B and g : B –> C. Prove: If go f is one-to-one, then fis…

A: A function h:M→N is said to be one-to-one if and only if hx=hy implies x=y for hx=hyx,y∈M. (1) Prove…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

A:

Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

A:

Q: 3. Prove if x is an isolated point of a set S, then x E bd S.

A: Since you have posted a multiple question according to guildlines I will solve first question(Q3)…

Q: Q3. Let z, y be real numbers such that ry 1 and z #y. Prove that

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.