Justify rigorously the given set is open in R³. (a) A = {(x, y, z) |x>0, y <0, x² + y² +2²> 9}. (b) B = {(x, y, z) | x² + y² < 25,0 < z < 25 − x² − y²}.

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A: In group theory, sets like Z12​ and Z18​ are examples of cyclic groups under addition modulo 12 and…

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Q: Q2) Let (A,T)be topological spaces; i=1,2,3. Then A, x A2 x Az is homomorphic to (A, x A2) x A3 I

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

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A: (a) Since f, using Mean value theroem w.r.t. x for two variables there exists a pR such that f(a +…

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Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

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Q: Dijec not need to prove you function is a bijection): (a) N and Z (b) Rand (1, ∞) (c) (0, 2) and (3,…

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A: Necessary Sufficient Nature f'x0=0 f'x0=f''x0=f'''x0=..=fn-1x0=0  fnx0&gt;0 n even Convex f'x0=0…

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A: We have to find the translation described in x,y→x+2,y-1.

Q: get the following sets {(x, y) : x ∈[1, 2], y ∈[1, 2]} and {(x, y) : x = 2, y ∈[0, 1]}

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Q: 20. Let V be an inner product space. Prove that (a) |||x|| - ||y||| ≤ ||x − y||. (b) (x, y) =…

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Q: Example 9. Classify each relation as constant, linear, quadratic, or neither. a) y = x² + 5x + 6 b)…

A: The solution is given below:

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Q: 16. Let vo = (a, b) and let v = (x, y). Describe the sets of points in R? satisfying the following…

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A: To Prove: x,y,≤ is not well ordered, where x,y is an open set with the usual order "≤".

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A: Answer= 8/9. The answer provided below has been developed in a clear step by step manner.

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Q: Determine whether the following sets are convex (1) {(x,y)=√x + √√]]l}

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A: As this is a multiple question type, only first question is to be solved, according to the Bartleby…

Q: 17. Use Theorem 4.7 to prove that Vx + 3 2 V3 + x/4 for

A: Let,

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Q: Let f(x) =x²-3X²- 24 x +26 a) the open intervals on which flx) is incheas in g aN the open intervals…

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