(ii) Show that the open set D= {z: Re z >0} is an open set. (iii) Sketch the set of points in C for which |z-1|<3. (iv) Find locus of points represented by z with z– 2 = 3.

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A: There are many types of relations Symmetric Asymmetric Reflexive Transitive

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Q: 1. Find the sets of interior points and boundary points of each set and classify the set as open,…

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Q: 1.2 Consider the following subintervals of the real line: A = [2,5], B = (0,4). %3D (a) Express the…

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Q: - Let D be a nonempty set and suppose that f: D→ R and g: D → R. Define the function f+g:D→R by (f…

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Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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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: يثبت The Cauchy-Schwarz inequality states that (f, g)| ≤ ||f||||g|| for every f, g ¤ H.

A: Using the property of inner product

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Q: 38. Let F be a closed set in R. Let y ¢ F. Show that there exists some ɛ > 0 such that the…

A: We have to prove the above statement.

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

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: 1) Describe and shade the portion of the x-y plane which is the domain and represent the domain and…

A: The domain of a function is the range of values that can be input in the function as the value of…

Q: Let A be a nonempty set and define B={kx : x∈A}, where k≥0 is fixed. Prove that supB=ksupA.

A: Given A be a nonempty set and define B=kx:x∈A where k≥0 is fixed. We have to prove that supB=ksupA…

Q: 4.4.4 List the relation on N defined bv h(x, y) e N x N : x > y } 9 Scanned with CS CamScanner as a…

A: The relation given is already a subset of the Cartesian product N x N. It can be written as follows.

Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

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Q: 1. Let F(x) = x. Compute the first five points on the orbit of 1/2.

A: Given:    F(x) = x2   First five points on the orbit of (1/2)

Q: 9. Describe and sketch the locus of z satisfying the condition |z – 1+ i| = |z – 2|

A: Answer

Q: If z, is a point on locus given by z+ 8 <2 and z, is a point on locus given by z+3|+ z-3| 10 then…

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Q: 2. Classify each of the following sets as open, closed, neither, or both. Then, find the closure. a)…

A: classify the given sets are open and closed.

Q: has

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

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Q: Which of the following sets are convex and which are not? 2 a. {(x₁, x₂) x² + x²₂² ≥ 3}. b. {(x₁,…

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

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Q: c) given an arbitrary convex function wcx) :R"→tR, prove fhat is a convex set.

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Q: 3. Let a, b e R such that a < b. Prove that the open interval (a, b) is an open set in R.

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Q: 1. List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a,…

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

Q: - Consider noncoplanar points A, B, C, and D. Using three points at a time (such as A, B, and C),…

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Q: Prove that the open ball B(p, 8) = {x : ||x – p|| < 8} is a convex set. [Hint: Use the Triangle…

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Q: R is not connected if T is * the indiscrete topology the trivial (usual) topology the finite closed…

A: Option 3 is correct.

Q: If T1 and T2 are two topologies on the same space X, then

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

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Q: prove that each of the following sets of points is collinear 1. A(4,5), B(3,3), C(1,-1) 2.…

A: As per guidelines, we are supposed to solve only first question. Please repost other questions…

Q: 3. Eliminate B and a from the relation x = Bcos(wt + a) in which w is a parameter not to be…

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Q: Sketch the set of all points (x, y) such that x + y| < e*.

A: Consider the function x+y≤ex. An absolute function is defined as x=x if x≥0-x if x&lt;0. So, there…

Q: 7.1.6 For the relation R = {(x, y) : x < y} defined on N, what is R-1, and what is the intersection…

A: The detailed solution is as follows below:

Q: 5. Let S be the open set consisting of all points z such that |z| < 1 or ]z – 2| < 1. State why S is…

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Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

A: Given that A is non empty set which is bounded below by 3. Given that B=x∈R|∃a∈A,b=1a The objective…

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A: The complete solutions is in given below

Q: b) Let C = {1, 2, 3, 4}. Find a relation R on C that has exactly 3 ordered pair members and is both…

A: Given C = {1, 2, 3, 4}. We find a relation R on C that has exactly 3 ordered pair members and  both…