Let kEZ". Let A be a Set svch that JA-RtI, Und RT aLA いt u fixed but uns pecified element of A). Define E人30 puo Yラ人人]にん Question: Show that the functhion f:X→ y defined by f(x):X U fa} ¥XEX is a bijection.

Q: Can you prove that it is injective by contrapositive? Also, how is f surjective when y exists in R?

A: Different equivalent definitions can be given for injectivity. The function f:A→B is injective if…

Q: 7. (10 points) Let f: A B be onto. Prove that S((C)) = C for all sets CC B.

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Q: Q2) Let (A,T)be topological spaces; i=1,2,3. Then A, x A2 x Az is homomorphic to (A, x A2) x A3 I

A: Definition : Let X and Y be spaces, and let B be the collection of sets U × V , where U ⊆ X and V ⊆…

Q: Conader the functon 22ty Sz,) = Among the folowng statements, select the ones that are true. Each…

A: using the basic definition of the limit of f(x,y).

Q: Operation o additien for all (XuX), (XXanXa) E TR. Define an operation fScalar multiplication…

A: consider a set S on ℝ2 where define an operation of addition on ℝ2 by X1,Y1*X2,Y2=X1+X2+1,Y1+Y2+1…

Q: C set strings of A and B. f: C→C is a function defined as f(c)=BcB any string c in C. a) Is…

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Q: Let A, B, C be sets. Let f : A → B and g : B → C be functions. a) Prove that if f and g are…

A: We shall solve both parts of this question that you have posted.

Q: 6. Determine whether the relation R on the set of all real numbers is reflexive, symmetric,…

A: Since you have posted a multi sub parts question according to guildlines I will solve first(a,b,c)…

Q: [Falkner Section 11 Exercise 26] Let A, B, and C be sets. Prove that if f: A → B and g: B→ C are…

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Q: to,' i.e. Let X be the space of integers and let R be relation 'less than or equal x Ry if x ≤ y…

A: The given relation is .

Q: Show that |qbu)|&x + 7元? the closed on interval [-元,T」 fo1 -π2 2くだ gluen that glven gux) for

A: Given g (x) = x +x² To find the maximum value of g(x). We can differentiate the function and equate…

Q: s paces Xand y, then both T and Ta boanded op erators are

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Q: 3)3 differant radical #'s rany is the set of all real #'s greater than o

A: Let us give three different examples of radical numbers in the next step.

Q: Show that if (xig)m.u. is re<rz , it is 8\x,r) C Blxirs] for Jx EX.

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Q: EX: Show that the telation ? >' is PartiaL oldering set of Por R-

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Q: on Give an crample of a dis continous bijection fo[at] = [1,2]

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Q: Show that day) = So, x** 0₁ X=Y. defines a metric on X₂

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Q: Question Show closed bale Y in Mebic space (X,d), Alsa ఎవక is set that ? then shaw is. complete d)is…

A: Given: Y is closed ball in metric space (X, d). To show:  1. Y is closed set in (X, d). 2. If (X, d)…

Q: Pove that 3-1 multiple of 2 for 1. is

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Q: 5)Let w-f(u, v), where u = X -y andv = a) Using the chain rule, express x-y in terms of x, y, of and…

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Q: be nonempty Subsets of Space X. Then Prove Let A and BJA inner product an TT 10) A CAL TT %3D (C) A…

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Q: Let Xis Hilber @ 1| + |1

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Q: Show that R is not compact.

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: Prove or disprove the following statement: A) Every T'3-space is T4-space?( explain your answer)

A: The statement "Every T3-space is a T4-space" is true. To understand why, let's first define the…

Q: A) Show that the surtace x-2yz ty' =4 is orthogon al to any of the sm tamily of sur faces…

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Q: Let A = {1, 3, 5, 7, 9, 11}, B= {a, b, c, d, e}, and ƒ be the function from A to B defined as below:…

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Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: State true or false with brief justification: L1[a,b] is a reflexive space

A: We use the Theorem: Let X be a separable normed space. If the dual space X* of X is non-separable,…

Q: What is true about the function *) - as *-* ? A) B)(x)→0 from below OC) f(x) is undefined OD)x)→…

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Q: Let F) be a differentiuble func Hen Such that Then g/2)= AS 912)= the equaben of Hhe fagat ine to…

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Q: 1) Let S be the equivalence relation on R defined by xSy iff x-y Q. Then a) [√3]-(-√3+r,req) b)…

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

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Q: 3) Let R be the set of all real numbers with the euclidean topology and f: R → R be continuous…

A: Yes, the image of a continuous function from a path-connected space is always path-connected.…

Q: 6.1 let T be a l'iner transformatio from Mga into M,a Sub that %3D

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Q: Show that the embedding of part (g) takes lines to "lines."

A: (h) We have to show that lines from ℝ2 maps to lines in P2. Lines in ℝ2 will be of the form ax+by=0.…

Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `