1. Let D = {(x, y) = R²|x² + y² ≤ 1} be the unit closed disc in R² and E := {(x, y) ≤ D\y = mx for some m € Q} (a) Are En S¹ and [0, 1] nQ homeomorphic? (b) Are E and D - E homeomorphic? (c) Are R² - E and R² – (D – E) homeomorphic?

Q: (a) Prove that f is a one-to-one correspondence. (b) Prove that ray AF is between rays AB and AE if…

A: (a) To prove that f is a one-to-one correspondence, we need to show that f is both injective and…

Q: 6. For any closed surface S, prove that || CurlF.N dS=0.

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Q: Let X=ℝ2 and define d,:ℝ2×ℝ2→ℝ d2((x1 ,y1),(x2,y)2)) = max{|x1 - x2|,|y1 - y2|}. Draw the…

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Q: 3. Let D = {(x, y, z) | x² + y² ≤ 1, z = 0.} C R³ be the unit disk in the xy-plane. Is D a Jordan…

A: A Jordan domain in the mathematical field of analysis is a subset of n-dimensional Euclidean space…

Q: Consider the closed and bounded set BCR2= [the ry-plane] defined by B = [0, 1] × [0, 1] = {(r, y) €…

A: note : Since you have posted multiple questions with multiple sub parts, we will provide the…

Q: = A: a) Let T be a linear functional from R³ to R. Show that there are 3 real a,b,c such that T((x,…

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Q: Show that the following subsets of C[0, 1] are isometric: X {fe C[0, 1]: f(1/2) >0} {fe C[0, 1]:…

A: Given :              X=f∈C0 , 1 : f12>0Y=f∈C0 , 1 : f12<0 Show that :  C0 , 1 and C0 , 2 are…

Q: 9.) Let E be the tetrahedron with vertices: (0,0,0), (3, 2,0), (0,3,0), (0, 0,2). E is bounded by…

A: The tetrahedron bounded by the planes              from the plane…

Q: Let V = C([−π, π]) be the space of continuous real-valued functions on the closed interval [−^,^],…

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Q: Are the following linear maps invertible? a) T: R3 → R³ defined by T(x, y, z) = (y, z, x)?

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Q: Theorem. Let D = = {(x, y) = R² : x² + y² < 1} be the closed unit disk, visualized below: dzumuoan…

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Q: 9.) Let E be the tetrahedron with vertices: (0,0,0), E is bounded by four planes: z = 0, x=0, 2 3…

A: Volume of tetrahedron

Q: Y Let V = {p(x) E P2(R) | p(-2) = 0} and W = E R³ | y– 2w +z = 0 be vector spaces. Consider the map…

A: Given V=px∈ℙ2ℝ| p-2=0 and W=ywz∈ℝ3|y-2w+z=0. It is also given that: B=-2-x, 4-x2 is a basis for Vand…

Q: Show that [0, o0) and [1, o0) are isometric.

A: We need to show that ,  [0 , ∞)  and  [1 , ∞) are isometric. We know that two metric spaces X and Y…

Q: . Let f(x+iy) = u(x, y) + iv(x, y), where u : R² → R and v : R² → R are defined by u(x, y) = x +…

A: A function is defined as a relation between a set of inputs having one output each. In simple words,…

Q: Consider the following. C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and…

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