b) Let X = R equipped with the standard metric. Then The boundary of [a, b),(a,b),[a,b), and(a, b) are all {a, b}

Q: Let r, y E R". Using the triangle inequality ||r + y|| < ||1|| + ||y||, show that r|| – ||y||| < ||r…

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Q: Consider the closed rectangle R = [a,b] × [c,d] in ℝ2 (under the Euclidean metric). i. Explain why…

A: Consider the closed rectangle R = [a, b] × [c, d] in ℝ2 (under the Euclidean metric). For (i), We…

Q: On the segment [0, 1) let's define metric with the formula: d(x, y) = 1 1 Is space ([0, 1), d)…

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Q: For the metric space (R, d) with d the usual Euclidean metric, which sets are in the topology T…

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Q: Exercise 4. If d is a metric on the set X, we define dcut : X × X → [0, 1] by dcut (x, y) = min(d(x,…

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Q: Define p: R^2 x R^2 to [0, infinity) by the rule p(x1,y1),(x2,y2)=squareroot((x1-x2)^2+(y1-y2)^2)…

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Q: Prove d(x,y) = 8 if x does not equal y and 0 if x equals y is a metric on R, the real numbers.

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Q: Let (x, y) and (z, w) be two points in R? state which of the following does not define metric on R?…

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Q: Let d : R? x R? – R, defined as d (x, y) = |21| wher e x = (x1, 2), y = (y1, 42), then d is metric…

A: The objective is to find the correct options.

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Q: Consider a 2-space with the following metric ds² = rdr² + ydy? (a) Calculate the nonvanishing…

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A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Derive the frenet frame, curvature and torison o the binormal indicatrix of hypocycloid?

A: We need to Derive the fernet frame, curvature and torsion o the binormal indicatrix of hypocycloid.

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Q: Let d: Rx R*→[0, ∞) such that d(XF)=√(x, y)² show that (R.d) metric space.

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Q: Let d: Rx R R, defined as d(r, y) = 0, then d is %3D O not metric because d not sats fied d (x, y) =…

A: The correct option is the first option. i.e.  Not a metric because d does not satisfy d(x,y) = 0 iff…

Q: If (x, y) is any point on the unit circle, and t is the distance from (1, 0) to (x, y) along the…

A: x, y is any point on the unit circle and t is the distance from 1, 0 to x, y along the circumference…

Q: Consider the sets N = R? \ {(0, y) E R² | |y| > 0}, L = {(x, y) E R² | |x| = |y|}, and let dr,…

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Q: Exercise. a) For every ,r, y E R, let p(r, y) =| x² – y? |. Is pa metric U..... on R? b) For every…

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Q: Exercise. a) For every ,x, y E R, let p(x, y) =| x? – y? |. Is p a metric on R? b) For every r, y E…

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Q: Let (x, y) and (z, w) be two points in R? state which of the following does not define metric on R?…

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Q: Let d: RxR-R, defined as d(z, y) = |r|, then d is metric O not metric because d not sats fied d (r,…

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Q: Draw the unit “circle” around the origin in the Euclidean metric, the taxicab metric, the hub…

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Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

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Q: Is the set S = {1,, 3, ...} with metric d(x, y) = |æ – y| for all x, y e S complete ? prove your…

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Q: d: R? x R? R Q2: Let d(a, b) = |b, - al + do(az, b2), where do denotes the discrete metric on R…

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Q: Let X = R? , and let d2 be the usual, Euclidean metric on X. Define another metric, d : X x X → R,…

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Q: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

A: Verify Green's theorem Where C is the boundary of the triangle formed by x=3 , y=0 , x2= 3y

Q: Show that w is in span(B) and find the coordinates of w with respect to B. B = 6.

A: Linear Algebra

Q: Let W = Span{u} where u -2. Let y = |1, find the distance from y to W L3]

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Q: _Show that the following are metric on R": d,(x,y) max, isn Ixi - vil : d,(x,y) E X-Yil.

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Q: Verify that (S, d) is a metric space. 10.2 Prove that d(x, y) < d'(x, y)s/2 d(x, y) for all points'…

A: To prove dx,y≤d'x,y≤2dx,y for all points x,y∈ℝ×ℝ where d is the usual metric and d' is the…

Q: Consider the sets N = R? \ {(0, y) E R² | |y| > 0}, %3D L = {(x, y) E R² | |¤| = |y|}, and let di,…

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Q: (a) Consider the metric space 0 (i) if r 1.

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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Q: 4. Let (R2, d) be the Euclidean metric, d(z, y) = V laı – 22 + \yn – y2l Describe and sketch the…

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Q: Let A be a subset of R with the usual metric such that A=A. If there exists r>0 such that d(x,y) <r…

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Q: Derive the frenet frame, curvature and torison of the binormal indicatrix of hypocycloid?

A: We need to Derive the fernet frame, curvature and torsion o the binormal indicatrix of hypocycloid.

Q: Let X = {a, b, c} Define d: X2[0, 0) by d(a, a) = 0, d(b, b) = 0, d(c, c) = 0, d(a, b) = 1, d(a, c)…

A: Metric Space: Let X be a set and a function d :X×X→ℝ Then X, d is a metric space if d satisfies the…

Q: Show that a indeed perojective plane of gives 0rder BİBD (n+n+,n+),!

A: We need to prove that a projective plane of order n indeed gives a BIBD (n2+n+1, n+1, 1)

Q: Calculate both sides of Green's Theorem when F = [-x²y, xy²] and y positively oriented circle x? +…

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Q: Let (X,d)be full m.J,P, defined as D:Xx X R,plxy=min{1, d , is the full metric on X, show.…

A: Metric Space: Let X be a set. If a function d :X×X→ℝ satisfies the following properties (1) dx, y≥0…

Q: Show that the sphere of radius 5, X(u, v) = (u, v, v25 – u? – v²) is regular.

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Q: Exercise 1.33. Let (X,d) be a metric spáce and d(x,y) 1+d(x,y)' d* (x,y) = Prove that the family of…

A: This is problem from my old works, where I used e instead of d*.