Exercise. a) For every ,r, y E R, let p(r, y) =| x² – y? |. Is pa metric U..... on R? b) For every x, y E [0, 0), let p(x, y) =| x2 – y² |. Is p a metric on [0, 00)?

Q: Q. Assume that (R, T) is the usual Euclidean topological space. D= {(x, y) E R?, y = x² +3} C R.Then…

A: Solution: Here the subset of ℝ2 is D=x,y∈ℝ2,  y=x2+3 Now, The set D contains points x,y which…

Q: 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x,y) Sr} Show that B,(r) is closed set…

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

Q: Show tuat a fuinchon fiR?>Rlefined by 3ں۔x5 ,ie (xxy]#Cojo) f cary)e je xcy=o Is continuous at (o,e)

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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: Let y e R. Determine if the following are metrics on R or not: d1(x,y) = sqrt(abs(x-y))…

A: A set X and a function d together called a metric space if the function d:X×X→ℝ satisfies the…

Q: Let (M, d) be a discrete metric space. Give explicitly a simplified expression for the following.…

A: Given: a) (M, d) is a discrete metric space. S1=Ba, 13, S2=Ba, 13, S3=Ba, 13 To find: Simplified…

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

A: TRUE

Q: non-empty set and d:X x X R de fined by d(x,y)=31 x- 41_ 2+|xl+lyl or disprove that d is a metric on…

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Q: What is the image of the vertical strip  {zϵC:−1<Re(z)<1} under e^z

A: A  conformal map is a one-one and onto analytic map. So, Re(z)=1 to a point. Here f1 is defined by,…

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: 1 Let C = [c1, c2, c3] = -1 2 3 -1

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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 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.

Q: Prove that the circle {(x, y), x² + y? = 1} is not homeomorphic to the interval [-1, 1).

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Q: 1. Let d, be a metries on X#o and let the function d: X x XR defined by: da, y) 1+ di(1, 9) Vr. y €…

A: 1.) As d( x, y ) = d1( x, y ) / 1 + d1( x, y ). So, d( x, y ) &gt;_ 0 for all d( x, y )  And d( x, y…

Q: The dimension of the set W = {(2t – s, s, t) : s, t e R} of R is equal to: %3D

A: The dimension of the set W={(2t-s,s,t) : s,t ∈ R} of R3 x=2t-s y=s z=t (t,s) ∈ R   This can be…

Q: By explicitly finding a ball centred on an arbitrary point and contained in the given set, show that…

A: The objective is to show each of the following sets are open ..

Q: In the adjoining figure, O is the centre of a circle. Two equal chords AB and CD intersect at P. If…

A: Given: In the above figure, O is the center of the circle and AB and CD are two equal chords…

Q: 5. Derive the distance formula between a point S and a plane through point P. |n PS d 3D |n|

A: Let Π be a plane passing through P with a normal vector n v is a vector parallel to n and passing…

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: Given (Xi,d;), i = 1,n as a metric space and we define the set X = II-1X; =X;*X2*X;*...Xn %3D Show…

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Q: Consider A = [3,7), B = (7,9] as a subset of R with the usual metric  dR (x,y) = abs(x-y). Show Ac…

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Q: 4. Let (R2, d) be the Euclidean metric, d(z, y) = V la1 – a2 + l - y2l Describe and sketch the open…

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Q: The norm of the set {cosh(i nn x},n E N, –-2<x<2 in the Hermitian sense is:

A: Given,                  cosi nπx, n∈ ℕ, -2≤x≤2

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: d(x,y) 1 - Y -

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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: Let X={9,10,11,..} and A= {10,11,12) and metric |defined on X as d(x,y) = |x-y Give the last answer

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Q: Exercise.3 Show that the following the functions d and p on X = R? are metrics or not, such that 2.…

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Q: A secant to two congruent circles, which is parallel to the line of centers OO', meets the first…

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Q: Given the usual metric d: RXR>[0, 0) defined by d(x, y) = |x- yl, write the set B(x, r) in interval…

A: Metric or Distance: Let X be any set A metric or distance is a function d : X×X→R such that  if we…

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

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Q: 3. Let r be the radius of the circle inscribed in a right triangle with legs a and b and hypotenuse…

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

A: Linear Algebra

Q: Prove Green's theorem for the rectangle defined by xo < x < x1 Yo < y < Y1 (a) directly, and (b)…

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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: 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: 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: Let d(x, y) = be a metric on R* then the open ball of center 2 and radius is: O ]1/2,1[ O None of…

A: I have used definition of open ball in metric space.

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 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: Let r be the inradius, R the circumradius, and s the semiperimeter of a right triangle AABC. Prove…

A: Given that, ∆ABC is a right triangle, r be the inradius, R be the circumradius and s is the semi…

Q: y (b, f (b)) (a, f (a))

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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*.