Show that the Euclidean metric is a metric( i.e. verify that the function d satisfies the three conditions in the definition of metric.

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: Prove that any contraction between metric spaces has at most one fixed point

A: According to the given information, it is required to prove that any contraction between metric…

Q: C. Show that a set A in R2 is open in the Euclidean metric A it is open in the Manhattan metric.…

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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: 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: State whether the following is true or false: a) The set [0,4] with the usual metric is…

A: (a) The set [0,4] with the usual metric is sequentially compact. This statement is true. Considering…

Q: メ=D¢ c omplex number %3D Doyou CGid) is metric space of not ??

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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: 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: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: Given metric is          d((x1, x2, x3), (y1, y2, y3)) = |x1 - y1| + |x2 - y2| + |x3 - y3|.   For…

Q: QUESTION 5 The set R-{0} with the usual metric is connected. O True O False

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Q: Show that d(x y) = √x-y defines a metric on the set of all real numbers

A: We will use the basic knowledge of metric spaces and set theory to answer this question.

Q: (a) Define a metric space (X, p). (b) Let p: R+ → R+ be a function defined by ln (2). Prove that p…

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Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

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Q: QUESTION 8 The set M={XEN:x<} with the usual metric is connected. O True O False

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Q: Verify that the discrete metric is actually a metric.

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Q: A metric space (X,d) such that the function d1(p, q) = (d(p, q)) i not a metric in X

A: Since (X, d) is a metric space. So, d is a function defined as d : X×X→R+∪0 and satisfy the…

Q: Let two metrics be bi-lipschitz, prove it is an equivalence relation

A: Find the attachment for the solution.

Q: Find all limit points of a set A = {(2+,2 – |n e N} on the Euclidean plane with the isual metric. Is…

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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: QUESTION 8 The set M= EN: X< with the usual metric is connected. O True O False

A: Since you have asked more than one question. So we will solve first one for you. If you want other…

Q: Every point in a discrete metric space forms an open ball.

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Q: Suppose E ⊂ Real Numbers. With respect to the Euclidean metric, is it possible that E' = { 1/n: n ∈…

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Q: The set R- {0} with the usual metric is connected.

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Q: Define new metric which is different from giving in above questions on X in question 01.

A: With the help of questions 01 we define a new metric on X

Q: d(x,y) 1 - Y -

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Q: The set [0,4] with the usual metric is sequentially compact.

A: We have to decide whether the given statement is true or false.

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

A: The unit circle around the origin in Euclidean metric is The unit circle around the origin in…

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: QUESTION 5 The set R/{O1 with the usual metric is not connected. O True O False

A: 5. TRUE The set, R - {0} with the usual metric can be written as the disjoint union of, (-infinity,…

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: Let a = (a, az); b = (b, ,b2) € R?, define the function %3D %3D d: R? x R? ---> R by d=(a, b) =…

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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: d (x, y) = (x - y)^2 does the function define a metric o

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Q: Example 3.16. In R with the Euclidean metric, the set [0, 1] is compact. However, note that with the…

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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: Show that by using an example that every metric is a pesudo metric but converse is not true.

A: To show that every metric is a pesudo metric but converse is not true.

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

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Q: C. Show that a set A in R2 is open in the Euclidean metric e it is open in the Manhattan metric.

A: The objective is to show that a set A in ℝ2 is open in the Euclidean metric⇔it is open in the…

Q: QUESTION 5 The set R-{0} with the usual metric is connected.

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Q: 5. Let h(P, Q) = min(xp, xQ) + min(yp, yQ). (Note: min(a, b) denotes the smaller of the two numbers…

A: We will use the definition of metric and answer all the other two questions accordingly.

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: the function d satisfies the three condition in the definitión of metric on page 39). 3. Show that…

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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: (X,d) be a metric space. Define f: X x X -> Real Numbers by f(x,y) = d(x,y) / 1+d(x,y) . show that f…

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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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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*.

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.