1. Draw sketches of the following subsets of C. (a) A = {z € C : |z+i|=1} (b) B = {z E C : |Zz+1| <2} (c) C = {z C : |zi|=|z2|} (d) D = {z EC: 2 ≤|z| < |z2| <4}

Q: 3. Find the maximum and minimum values of a 1 āz - on |z| ≤ 1 (where a with la| < 1 is fixed)

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Q: 13. Let U be the set of real numbers, A = {x |x is a solution of x - 1 = 0}, and B ={-1, 4}.…

A: Let's find.

Q: 2/ If (xit) is top spaces.t x+y there are two disjoint Pen sets containing them Prove {y? y ex…

A: We use the following theorem  Arbitrary union of an Open set is open.

Q: 5. In which annular domain is f(z) = (a) A(0,0,4) (b) A(0,0,2√2) (a) (b) 1 24 - 16 (d) holomorphic?…

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Q: (-х+а, х < —1 If f(x) = {x² + 5, –1 < x < 2 is continuous then х — b, х 2 2 a + b =?

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Q: 2. Let f(x) = x - 5 and g(x) = x² +3x. a) Find (f + g)(x) b) Find (f – g)(x) c) Find (f g)(x) d)…

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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: 4. Consider the set of functions S = {-3(x² – 4)°,7|x² – 4|³}. a. Show that S is a linearly…

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Q: 22. Show that if F₁ and F₂ are primitives for f: G → C and G is connected then there is a constant c…

A: Given : F1 and F2 be the primitives of f : G→ℂ, where G is connected. To show : There is a constant…

Q: (e) For the set W = {[1], [3], [13]} ≤ Z18 determine the set C = h¯¹(W). Then for the set C C Z12…

A: Modulo function

Q: Show that A = (aj) = Cmxn. Then || A||2 ≤ || A||F ≤ √ Rank(A)|| A||2 V

A: Matrix A=(aij)∈Cm×n: This notation indicates that A is a matrix with complex entries, where m and n…

Q: d. Consider the following convex optimization problem min fo(x) s.t. x EN where O C R" is a convex…

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Q: 8. Use finite differences method to classify the following relations as linear, quadratic or…

A: In this question, the concept of the finite differences method is applied. Finite Difference Method…

Q: positive real numbers such tha ith proof the minimum value of

A: Given, x,y,z be a positive real number such that x4+y4+z4=1.The minimum value of…

Q: 2. (a) Let A = {x = [0, 1]: x has a decimal representation not containing the digit 5.} Show that A…

A: We shall use certain results from measure theory

Q: (b) Show that the set A = {(x, y) E R² : y > 0} is not complete.

A: Given: The given set ,

Q: Let S = (0, 0), T = (-∞, 2], and W = [-4, 2). For each intersection or union, choose the correct…

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Q: 10. Determine whether each of these functions from {a, b, c, d} to itself is one-to-one. a) f(a) =…

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Q: Assumed that F OG = Ø, show that we can express the infimum and supremum of F NG in terms of the…

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Q: The value of the Wronskian of set of functions {x² - 3x +1, 2x² - 6x + 2} is O a. 13x O b. 7x? 2x? O…

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Q: A = { 3, 0, 1 } B = { y, x }  what is the cardinality of A x P(B) ?

A: Given that, A = {3,0,1}  B ={ y , x }  Power set of B = {{},{x},{y},{x,y}}  Generally cardinality is…

Q: (2.1) Consider the following topologies on X = {a, b, c, d} and Y = {r, y, z, w} TX = {X,0, {a} ,…

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Q: . Let A CQ satisfying the following conditions: (a) A + Ø and A # Q. (b) If p E A and q E Q with q <…

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Q: (2) Prove that for all x, y € R, we have 8xy³ ≤ (2x + y³)².

A: A.M&gt;=G.M

Q: Let X = {1, (b) xm} and Y = {y₁, Yn} be finite sets. Determine the number of one to one functions…

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Q: Let A, B C X be subsets and f : X → Y a function. Show that ƒ(AÑ B) ≤ f(A)Ñ ƒ(B)

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Q: 2. What are the Domain, the Codomain, and the Range of the following functions. •f:R→R f(x) = x² + 2…

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Q: 63. Given n nonzero numbers o1,...,on, show that the minimum value of S(x1....,xn) = xfo7 +..+ x…

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Q: 15. Show that if f: A- 1 f(GnH)=f(G) B and G, H are subsets of B, then f-¹(GUH)=f(G) Uf (H) and nf…

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Q: 1 Expand f (z) = in the domain 4 < |z| < 12 (z-4)(z-12)'

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Q: 7. Sketch the set {(x, y) = R² : ||(x, y) || = 1} when: (a) ||(x, y)=√√√x² + y² (b) ||(x, y) || =…

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Q: 4. Let X an Y be sets with A, B C X and let f: X→Y be a function. Recall that {f(x) EYE A}, and…

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