Let d be a trivial metric on a non-empty set X and d₁: XXX → R such that x = y d₁(x,y) = { 2 if x #y' Show that d&d, are equivalent? vx, y EX

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: A) Let (R, T) be the usual topological space and f: R → R s.t. f(x) = 2x + 2.Show that f is…

A: Given that (ℝ,τ) is the usual topological space (the standard euclidean topology over ℝ). Also given…

Q: Let f(x.y) = ry. a. Sketch/describe the domain of g Domain = { (x,y)]yz0} b. Sketch a contour map…

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Q: Let (R,d) be diserete metric space then R is not compact . True or false.??

A: Let (R,d) be diserete metric space then R is not compact . True or false.??

Q: Let f: X → Y and g : Y → Z be functions. (a) Show that if g of is one-to-one (or "injective"), then…

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Q: Let f be a function of two variables such that Vf(x, y) and f(2,1) 4 = af • If, in addition, x =…

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Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Let u = and v = Show that is in Span (u, v} for all x and y. y -1 5

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Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

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Q: »[32° dx = 3x² A) 6 B) 7 C)8 D) 9 타 10 xdx A) - B) - Suppose F and G are continuous functions with…

A: We have to answer the following...

Q: (R,t) usuall top space and A=(0,1),B = (a,b); a<b. prove A=~B ??

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Q: O Let ( X, T) be topdogical space and A,B CX- ldentify the Condition when a (Int(A)u FrlB) Ul F CA)…

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Q: #4. Set XyZ F(x4a)= | 4 3ゃex xy tyz+zX メ*ytを Show that F is 11 an open set ontaining (1,0,-1). in O…

A: Given - Let Fx, y, z = xyzxy + yz + zxx + y + z To Show -  Show that F is 1 : 1 in an open set…

Q: 3: Let a function d: R* x R* → R defined by d(x, y) = E-\xi - yıl Vx = (x1,...,x4), y = (y1,...,y4)…

A: Here we will prove that d is a metric function.

Q: The largest set on which the function f(x, y) = 1/(2-x² - y2) is continuous is A. The exterior of…

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Q: Show that close ball Y in a metfic space (X,d) is a closed set also show that if (X,d) is complete…

A: Show that closed ball Y in a metric space (X,d) is a closed set also show that if (X,d) is complete…

Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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Q: If f is a continuous function satisfying f(x) = x* - x+(1-x)[,f'f)dt + x[ (t -r°)f(@dt for all x,…

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Q: Forz € C, define the linear operator T₂ : C → C by T₂(w) = W. Characterize those z for which Tz,…

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Q: 3. If x and y are two n-vectors, then prove that 1 (a) |x*y|< ]], \»l, - +=1 (Holder inequality) 1…

A: Given that x and y are two n-vectors. We have to prove the following :                          (a)…

Q: (1) Using the second order Runge-Kutta method (Modified Euler's method), plot the solution of the…

A: y'=10(y-y2)    0≤t≤5,     h=0.01,     y(0)=0.5 y1p=y0+hf0 f0=F(t0,y0) t=0, y0=0.5 f(x,y)=10(y-y2)…

Q: Let P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

Q: dual of c, is isometric to l'.

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Q: Show that if (x.g) m. is re<rz, it is 8lx,r) C Blx,g) for x Ex. U.

A: Metric or Distance: A metric or distance is a function ρ :X×X→ℝ such that ρx, y represents the…

Q: If T is surjective and V = span(3), prove that W = span(T(3)). Hint: given that span(T(3)) ≤ W, it…

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Q: Theorem (Axioms of a Derived set): Let (M,d) be a metric space and let S,T C M. Then (1) S CT S' S…

A: Note: Our guidelines we are supposed to answer only three subpart . Kindly repost other subpart as…

Q: For each of the sets find the set of boundary points and the set of interior points. Also state if…

A: We have, (a) [0,1]U{4} (b) x,y∈ℝ2:x2+y2=1 Here we have to find the  o find: Interior Boundary…

Q: 1. Consider the following three domains in the space: {(x,y,z) E R³:1 ≤|x+y+z<4}, {(x,y,z) € R³ : 2…

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Q: Let (X, d₁) and (Y, d₁) be metric spaces. Let f: X Y be continuous function, then f-¹(G) is open in…

A: Solution