Suppose Show that p: X²> t0,00) defined by d is a metric n X, 2. 3d (x,y) 2+3d (xy) metric an p Cx,y)= is also a X. १S

Q: Let M = {0,1} and d be the discrete metric, then d is not complete %3D O True O False

A: Discrete metric space.

Q: „Consider the space (R,T). where t={R,Ø}U{ACR:[0,1)CA} :Then (R,T) is not T, O T, but not T, O T2 O…

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Q: 1. Suppose that (X, dx) and (Y, dy) are metric spaces and f: X →Y is a function. For a, b e X,…

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Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

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

A: Wr Know that,"Let X = (-∞, ∞) and C consists of all intervals of the form (a,b).The arbitary union…

Q: where a, b e X, is also a metric on X. 2- Let (X, d) be a metric space and T the corresponding…

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Q: The Cartesian product X = X1 x X2 of two metric spaces (X1, d1) and (X2, d2) can be made into metric…

A: 4) To define a new metric on X which is different from the metrics mentioned in (1), (2), and (3).…

Q: Let X = [0, 1] and d be the usual metric, d(x, y) = |x- y| %3D O True O False

A: TRUE

Q: Let d be a metric on X + 0. Show that the function e defined by d(a, b) 1+ d(a, b) e(a, b) = %3D…

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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: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: Write a homeomorphism between the unit closed disk A = {(r,y E R2 : a2 + y? < 1)} and the unit…

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Q: Q2: (a) Let M = {x,y,z) and define d: MXM → R by Determine whether d is a metric on M. d(x,x) = d(y,…

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Q: Let d: R x R R. defined as d (z, y) = |#1 - ml wherez= (21,2), y= (y.), then d is %3D Osemi metric…

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Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

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Q: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

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Q: Show that the closed interval [0, 1] and the unit circle S' = {(x, y)|x² + y² = 1} cannot be…

A: Suppose there exist a homeomorphism f:[0, ∞)-----&gt;R then restriction of f f|[0,∞)\{0} : [0,…

Q: (f) The set B = {(x, y) E R² : x² + y² < 1} as a subset of R? with indiscrete metric is…

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Q: On a previous homework, you proved a Bolzano-Weirstrass theorem for R3 with the metric d((x1, X2,…

A: Given a metric in ℝ3 and given it satisfy Bolzano-Weirstrass theorem. We need to prove that if a…

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: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

Q: In a metric space (X,p), if lim un = u, lim v, = v, show that lim d(un, Vn) = d(u, v). %3! n-00

A: Convergent Sequence: A sequence xn in a metric space X, d is said to converge to a point x∈X if for…

Q: evaluate $ A. dš over the sphere R = 3 and show whether Divergence theorem holds or not. %3D

A: Calculate surface integral over R=3 or dr=0 ∮A→×ds→=∮R cos2θ,π,0×r2sinθdθdϕ,rdϕdr,rsinθdrdθ…

Q: Suppose d is Show that p: X?> [0,00) defined by a metric on X, p Cxy)= 3d (x,y) %3D 2+3d(x, u) is…

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Q: let M= To,2] anel d be bhe usual metric then d is not compact or Compact ?

A: A set is said to be compact if and only if it is closed and bounded.

Q: i) Pe(x) = 0 iff r € Ë. ii) Prove that pe is uniformly continuous.

A: i Let E is the non-empty subset of a metric space X,d the distance from x to E is defined as…

Q: Is the set S = [0,1] with the discrete metric d separable? Explain

A: We Know that a set is separable in metric space only if it contains a countable dense subset in it.…

Q: 4. Let {Xn, dn fnɛN be a collection of metrič spaces of D(x, y) = sup{d,(xn, Yn)/n | n E N} a metric…

A: Here we show positivity, symmetric and triangle inequality one by one.

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: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

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Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+ z) = d(x,y), x,y € N…

A: Answer

Q: Given the discrete metric space for n E Na the discrete metric o defined by the distance function 0…

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Q: If (X, d) is any metric space, show that another metric on X is defined by d(ry) d(r, y) 1+ d(x, y)…

A: The given problem is to show that X is bounded in new metric d(x,y)  Already given (X,d) is the…

Q: metric

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Q: Suppose h: X –→ Y is a homeomorphism and suppose d is a metric on X so that X is complete with…

A: a) In the question it is given that suppose  h:X→Y is a homeomorphism and  d:X×X→(0,∞)  is  a metric…

Q: If X =R and d(x,y)= 1, then X not metric semi metric metric space

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Q: Show that it A is subset of X any with the discre te metric then %3D

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Q: Let (X, d) be a metric space. Define dˆ : X × X → R, by: ˆ
d(x, y) = min{1, d(x, y)}. (a) Prove that…

A: To verify that d^ satisfies the axioms of a distance function

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…

Q: close ball is an open

A: Consider the metric space X,d, where d is the usual metric on ℝ4. Bra is the closed ball with centre…

Q: For any non empty set X, the discrete metro d:  X^2 to {0,1} is defined to by  d(x,y)={0, x=y and…

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Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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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: Let X=ℝ2 and define d2,:ℝ2×ℝ2→ℝ by d2((x1 ,y1),(x2,y2)) = max{|x1 - x2|,|y1 - y2|}. a) Verify that…

A: Given that X=ℝ2 and defined by d2:ℝ2×ℝ2→ℝ Part(a): Now from given d2:ℝ2×ℝ2→ℝ by…

Q: Let f: (N,d) (N, d), where d is the usual metric and di is the discrete metric, defined f as f (x)…

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Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.