1. Sketch the following sets and determine which are domains: (a) \z – 2+ i|< 1; (c) Imz > 1; (e) 0 < arg z 4; (d) Im z = 1; (f) \z – 4| > ]z].

Q: Solve the following inequality graphically in two-dimensional plane:y &lt; – 2

A: We first draw y=-2 line

Q: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

A:

Q: 2 Let F be a closed set for k = 1, 2, ... , n in (S, d). Show that |J F, is closed. k=1

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: 7. Let x E R and A be a nonempty subset of R. If x is not a limit point of A then x is an isolated…

A:

Q: Consider the reglon bounded by the graphs of y- and y- VE

A:

Q: 2.4 (Ross, 2003, page 593) Show that the relationship $ (x) = x;¢(1;, x) + (1 – x;)ø(0¡, x) is true…

A:

Q: 6. Compute the norms for (a)||f()||, (b) ||F(x)||2, (c)||f(x)||, where f(x) = -V1-r", defined on [0,…

A: Given

Q: 3;) Find the välie of k isó that. the furnictioin %3D .kx*. .if. x 2 2. is. continuous .cvery iw…

A:

Q: In Exercises 1-7, (X,M) is a measurable space. 1. Let f : X R and Y = f-1(R). Then f is measurable…

A: Given: Let's consider X,M is a measurable space.  Here f:X→ℝ¯ and Y=f-1ℝ

Q: 7. Suppose that X and Y are disjoint sets and k E Z>o. Prove that the function k so:U(P:(X) × x…

A: Let us solve step by step.... X and Y are disjoint sets...

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

A:

Q: Prove that ', 12 and lº norms satisfy the properties (N1)-(N3). (N1) (Positivity) ||x|| > 0, and ||x…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 1) If the rational preferences of a consumer over bundles of (x₁, x₂) satisfies which of the bundles…

A: Solution of the problem as follows

Q: 2. !. (n!)². > Show that there is no holomorphic function f (z) on any open set containing 0 such…

A: To Prove: There is no holomorphic function fz on any open set containing 0 such that fn0=n!2.

Q: find the poin ts of discontinuity of the following func tion s : (a) g (x,y) %3D (b) f laiy)

A: g(x,y) = 11-x2+y2The point of discontinuity is where g(x,y) approaches ∞. Or the denominator of RHS…

Q: 2/Define the topology If X-R² and T = {GK, KERZULAR} GK = {(XY) ER²|x+y >K3 Prove is topolgy on R²

A:

Q: Prove that the set {x e R : 10/x - x > 0} is bounded. Prove that the set {x E R : x² – 25x > 0} is…

A: To prove that the set x∈R:10x−x&gt;0 is bounded. To prove that the set x∈R:x2−25x&gt;0 is unbounded.

Q: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

Q: Suppose that fa f (uniformly) and gn= g (uniformly) (pointwise) on A CR. on a set ACR. Show that…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Show that there exists a set N C (0, 1) that is not Lebesgue measurable.

A:

Q: 2. Find three positive integers x, y, and z if their product is 27 and the sum is a minimum.

A: Given three positive numbers are  x, y, and z Product = xyz = 27 And sum is minimum

Q: Show that X and Y are independent iff E(f o X – go Y)² E(f o X – E(f o X))² for all Borel-measurable…

A:

Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

Q: Consider the subset W = {x € Q : xª > 1} of Q. 2.1 Express W using interval notation. 2.2 Is W…

A: W = x∈ℚ: x4&gt;1 This implies that W is a set of rational numbers 'x', such that x4&gt;1. if x4…

Q: 2.5 For each of the following properties, give an example consisting of two subsets X, Y CR', both…

A:

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: O Suppose that ACR and A is bounded below with inf(A) = L> 0. Prove that for a > 0 and b > 0, the…

A: Since the posted question has multiple questions but according to guidelines we will solve the first…

Q: 2- Find the set of lower bounds and the set of upper bounds of the following sets in Q, if it…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Theorem 3. Every solution of Eq.(1) is bounded if a + < 1.

A: As per the information, we will first find a recurrence relationship of the sequence, then find its…

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

A:

Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

Q: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x&gt;2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

Q: Let S be a nonempty set of real numbers that is bounded from above and let x = sup S. Prove that…

A:

Q: (7) First define a semi-finite mea sure. Then prove that if u is a semi-finite measure and (E)- then…

A: The method is given below with proper explanation and definition: Let (X,Σ,μ) be a measure space.…

Q: has

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

Q: Examine the continuity of if 0 a at

A: We need to examine the continuity of the given function f(x) at x=a.

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

A:

Q: 4. Use the least upper bound property of R to prove that every nonempty subset of R that is bounded…

A: Solution

Q: 3. DETERMINE THE UOLUME OF THE REGION BOUN DED ABOUE BY THE PARABOLOID Z=2-x²-y BY THE CON E

A:

Q: 6. The distance between a point x ER and a nonempty set A CR is defined by d(x, A) = inf {\x – y| :…

A: In this question, we have to prove that d(x,A)=0 iff x belong in closer(A)

Q: Solve the following inequality graphically in two-dimensional plane:y + 8 ≥ 2x

A: Given:

Q: 1) 1 let X = {i,Q,3,4} woith topology T- {0,{17u3} f64}{4},x} What What the closed sets the of…

A:

Q: 1 : Determine the manimom and Nalve of the function minimumn x'tyス condition under the *+ 2 7?== 1.

A: To find the absolute maximum and minimum of the function using lagrange multiplier

Q: 4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = 0.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: The relation xy >= 1 where x,y are on the set of all integers is reflexive. True False

A: Given relation xy≥1, where x,y are on the set of all integers

Q: 4.Describe the largest set S on which it is correct to say that f is continuous S (x,y) = In(1–x² –…

A: Given problem:- Describe the largest set S on which it is correct to say that f is continuous C.…

Q: Q1- If f is bounded and define on [a, b] then show that m(b – a) < L(P,f)< U(P,f) < M(b – a). Q2-…

A:

Q: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

A: