2. For points p = (z1,Yı) and q = (12, 42) € R², the taxicab metric is given by d(p, q) = |r2 - 1|+ \92 – y1|- Verify that this is a valid metric by showing that it satisfies all four necessary criteria of a 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: 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: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

A:

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

A:

Q: 1. Using Euler's equation, show that the shortest distance between two points (r1, yn, z1) and (r2,…

A: Given, the points are (x1, y1, z1)   and  (x2, y2, z2).

Q: Show without using the Heine-Borel theorem that the line R is not a compact set in R ^ 2

A: The proof is done using the definition of compact set. A set S is said to be compact if and only if…

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

A: TRUE

Q: Find the distance (in appropriate units) between the points P(2,−1)P(2,−1)and Q(3,4)Q(3,4)in the…

A:

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

A: Assume that A is open subset of R2  in an Euclidean metric. So A can be written as,…

Q: Find the curvature k of the circle C : (x – 2)² + (y + 3)² - at the point (5, 1).

A:

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)…

A:

Q: 10) Define the metric d on R² by d (x₁, y₁), (x2, y2)) = max{|x₁ - x₂], ly₁ - yal}. Verify that this…

A: This is a question of topology.

Q: Let M = 0,2) and d be the usunl metric, then dis not complete O True O False

A: False

Q: 24. Consider the lines l: x = Xo +tv and m: x = x1 + su. Show that l and m intersect if and only if…

A: We will prove the given statement.

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

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

Q: The set [0, 1] with the discrete metric is compact. True False

A:

Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

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, ∞)----->R then restriction of f f|[0,∞)\{0} : [0,…

Q: 5. If мая abstract manifolds, PEM. f a) Define TpM and dfp, where M²₂N is smooth map beton by dfp:…

A:

Q: A(n)________ is the collection of points in the plane the difference of whose distances from two…

A: The given statement is "A(n)________ is the collection of points in the plane the difference of…

Q: Let X = R, and Let M = [0, 3] and d be the discrete metric, then M is

A: M=[0,3] is compact ,since it is closed and bounded in real R.

Q: Verify Green's theorem in the plane for M = -2y and N = x , where C is the semi circle x2 + y? = 9,…

A:

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…

A:

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

A:

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

A:

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 -

A:

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: Find the curvature k of the circle C:(x – 2)2 + (y +3) = 25 - at the point (5, 1).

A: circle equation is x-22+y+32=25 and point 5,1find curvature k

Q: 2. Use analytic geometry to verify that quadrilateral JKLM is a parallelogran 1C). YA K(-1,1) 2 -2 0…

A:

Q: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

A:

Q: Evaluate f 3z2dz over C, where C is the upper half circle |z – connects points O and 1+i. -글히 %3D플…

A:

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

A:

Q: Find all limit points of a set A = {(2+,2 – 1)|n e N} on the Euclidean plane with the usual metric.…

A:

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

A:

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

A:

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: (a) Prove that curves a, ß: I → R² are congruent if k̟ = Kg and they have the same speed.

A: Part (a):   If any two continuous function defined on the interval I, and the two curves are…

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

A:

Q: Calculate the metric coefycient h2 of the coordinate system defined by 503 x=5q1 +3q2-53 1243 y =…

A: The coordinate system is given by x=5q1+3q2-5q3y=2q1-18q2-12q3z=3q1+11q2-q3 Here, x1=x, x2=y, x3=z…

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

A:

Q: Q5. Prove that the circle {(r, y), x² + y? = 1} is not homeomorphic to the interval [-1, 1].

A: Here we can prove this question using topological property connectedness.

Q: The set E={xER:xs2x² – 1} with the usual metric is compact.

A:

Q: 2. Does d(x, y)= (x-y)? define a metric on the set of all real numbers?

A:

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

A:

Q: Let C be the circle |z| = 4 traversed once in the positive sense (counter %3D clockwise). Compute :…

A:

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

A: