Show whether the sets are connected or not. 1. {z € C | 1< ]z| < 2} 2. D(0, 1) U D(2i, }),

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

A: 3.3 is also known as Hausdorff Property. d(x1, x2) &gt; 0 for every x1 ≠ x2   (metric space…

Q: (b) Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or…

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Q: 1.* Let U C R? be an open and bounded set and u e C²(U) n C(U) satisfy -1 Ди(х) : in U. 1 + |x|

A: Since I have solved similar ones, Let us show that u attains its minimum on the boundary

Q: A set S c C is called bounded if there is a positive number M > 0 s.t. |2| 0} (c) {z = x + yi, |z +…

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

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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: State whether each of the following sets is open, closed, or neither. Which are compact? (You do not…

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Q: Let S = { all (x, y) with æ and y rational numbers j. (a) What is the interior of S ? (b) What is…

A: We know interior of the set of rational numbers is empty and boundary of the set of rational numbers…

Q: Prove and show complete solution: Let ƒ be defined on an open interval (a, b) and assume that for…

A: please comment if you need any clarfification. If you find my answer useful put thumbs up. Thank…

Q: 5. Which of the following sets S are polyhedra? If possible, express S in the form S = {x | Ax 3 b,…

A: Given set is S=x∈ℝn:x≥0, 1Tx=1, ∑i=1nxiai=b1,∑i=1nxia2i=b2, ai∈ℝ, b1,b2∈ℝ Clearly S is defined by…

Q: Given sets X and Y. For x EX and for y E Y, we form the Cartesian produc X x Y := {(x, y) : x E X, y…

A: Given that: X={-1, 1}Y={-2, 0, 2}

Q: 1. Sketch the following sets and determine which are domains: (a) \z – 2+ i| 1; (e) 0 4; (d) Im z =…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. ореn b. closed c. clopen d. is…

A: Let (X, T) be a discrete space.

Q: 10. Let ro = all points (x, y, z) for which (xo, yo, zo) and r = (x, y, z). Describe the set of %3D…

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

Q: Write one or more inequalities that describe the set of points. The closed region bounded by the…

A: For closed region , Sign of equality holds. Equation of sphere is -  x2+y2+z2=a2 ,where a is the…

Q: 8. Sketch the sets X = {(x,y) E R :x +y s 1} and Y = {(x, y) ER:-1sys0} on R. On separate drawings,…

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Q: 15. Given nonempty-subsets of R², say Y1,..., Yn, let Y* = {E÷=1 Y; | Y1 € Y1,.., Yn E Yn}. Given p…

A: Given: Y*=∑j=1nyj | y1∈Y1, y2∈Y2, ..., yn∈Yn and, for any p∈R2 and nonempty subset Y⊂R2, vp,…

Q: Consider the sets X = {(x, y) E R²: x² + y² ≤ 1) and Y = ((x, y) = R²: x ≥ 0]. Sketch on R²the sets…

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Q: Q2: In Euclidean metric space (R², 1. 1), is the set A = {(x, y): xy < 1} compact? giving the…

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Q: Q.2. Let t, be the particular point topology on X (a, b, c, d), and f:(X,T)-(X, r.) is defined by…

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Q: 1. By using the standard metrie X. Determine JA and X\A. Justify your answer. ソour State whether A…

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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: Proof by contrapositive that for x, y ∈ Z, if xy is even then either x or y is even.

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Q: A set S C C is called bounded if there is a positive number M > 0 s.t. |z| 0} (c) {z = x + yi,…

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

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Q: and for the following situation Determine forulas for w=(z,y, ) 2=z (t), y=y(w, v, p), z==(v,p),…

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Q: IF f:x-у and vCy proof fy/v)=x/f"(v) and proof f is continous iff f'(E) is closed set in x for every…

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Q: Suppose that a > 0 is a positive integer, and n > a. How many lattice paths are there from (0, 0) to…

A: According to the given information, suppose that a&gt;0 and n&gt;a. It is required to find the…

Q: PO しl Let 4Let R be the reaion enclosed by oe en closed by Jhe graphs [sreph R ont omt y=8; x= o ind…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: When is S \ T = T \ S for sets S, T?

A: Given that two sets S and T. The definition of S\T=a∈S,a∉T and T\S=b∈T,b∉S. Here, neither of these…

Q: Let X={9,10,11,..} and A= {10,11,12) and metric |defined on X as d(x,y) = |x-y Give the last answer

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

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Q: O Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or x = y…

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Q: 8.Prove the following relationships: a) I(X,Y)=H(X)-HOX/Y)

A: Solution

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

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Q: VII.1. Describe the set (R × Z) ∩ (Z × R)

A: We have to describe the set (R × Z) ∩ (Z × R).

Q: -[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…

A: Given- Let S be a nonempty set in Rn and let x¯ ∈ S. Consider the set C = y : y = λx - x¯, λ ≥ 0, x…

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: {S := n(x, y) | 2x + 3y ≥ 12}

A: First we draw the line then find the region.

Q: - Sketch the set S = {z: |z − 2 − i|≤2}. Find the interior, exterior and boundary set of S. Then,…

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Q: Give all boundary points of D.  Is D open, closed or bounded? Justify all your answers.

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Q: 1. Let X = {1,2,3} with the topology 1={X,ø,{1,2},{2,3},{2}}. Verify whether X is regular or not.…

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Q: (c) If A E M,(R) is non singular and n 2 2, prove that det(adjA) = (det A)"- (d) If A E M,(R)…

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Q: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

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Q: Oct 31, 2021, 16:44 Problem-2 0 Let foo = x?-2a and gcauz VI-x 6) is q CG (-Z)) defined! domain ind…

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Q: 1. Prove algebraically that for any sets X and Y, we have Y – (Y – X) = X nY.

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

Q: Let S be some non-empty collection of real numbers. We know that Vx E S, 3y E S such that x2 y.…

A: In this type of question you need to assume the set of x,y and S that satisfy the condition given in…

Q: Q3- Give an example of two denes not equal X subsets of (R,Te) and (z,ror ). ( Explain your answer…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.