Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x + y = 0. b) x = ±y. c) x − y is a rational number.

Q: (b) Consider the set R. Prove: (i) VxER, x 0 = 0; (ii) If x y = 0 then either x = 0 or y = 0, or…

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Q: Prove that if (R,+, ·) and (R',+', ') are isomorphic integral domains, then their ficlds of…

A: It is given that Two isomorphic integral domains

Q: Let R be the relation on R defined by R= {(r, y) ERXR|r² + y? 4 or r y}. Answer each of the…

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Q: Let f : R R with f(x) = -2x +5 Prove that f is onto.

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Q: A point y in space either belongs to a given closed c ough y so that whole of the x lies in one open…

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Q: Prove that L∞[a, b] is complete with respect to the norm || · ||∞.

A: The elements of L∞[a,b] are the essentially bounded measurable functions.

Q: 5. Suppose that f: X →Y and g : Y → Z are bijections. Prove that gof is a bijection and that (go f)-…

A: We will find out the required solution.

Q: Suppose M and N are C manifolds, U an is C. Show that there exists a neighborhoo that F can be…

A: Let us start solving the problem step by step

Q: Let X,Y,Z be sets and f:X→Y, g:Y→Z. Prove that if g is injective and g◦f is surjective,then f is…

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Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: The relation R is defined on R^2 by (x1, y1) ∼ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1) and…

A: a. The relation R is defined on R^2 by (x1, y1) ~ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1)…

Q: Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of…

A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: Is the relation R, defined on the set of real numbers by xRy if 2x + y = 3. antisymmetric?

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Q: The relation R {(x, y): 5x +5y > 21} on the set of positive integers is: %3D alyyall a is None of…

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Q: (d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by (i) connectedness, (ii)…

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Q: Let T be the cuclidean topology on R and r' be the finite-closed topology on R. Define f : (R, 7') →…

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Q: Let L be the set of all lines in a plane and R be the relation in L defined asR = {(L1, L2) : L1 is…

A: The given relation R is defined as follows.

Q: A. Let H be the set of all points (x, y) in R? such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Define a bijection function f : N → {n : n > 10, n e N}, and prove it.

A: We have to define a bijection function f:N→n:n≥10,n∈N and prove it. Define a function…

Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: A. Let H be the set of all points (x, y) in R2 such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Let R be equipped with the standard topology. Define q : R? → R, q((x1, x2) : Jx1, if x1 2 x2 |X2,…

A: Question: The function q:ℝ2→ℝ defined by qx1,x2=x1, if x1≥x2x2, if x1<x2 where ℝ2 and ℝ are…

Q: Prove that R* ⨁ R* is not isomorphic to C*.

A: First, we count the elements of finite order; R has two elements of finite order that is (1, -1) C…

Q: Consider the relation S on N defined by uSy⟺5|(y+u). Is it reflexive?

A: No it is not a reflexive relation

Q: Show that the map f(z) = 1/z is a bijective holomorphic map from S2 to itself, using the charts for…

A: Given: f(z)=1z

Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

A: Given that T be the co-finite topology on ℝ and d0x,y=x-y To prove that the function f:ℝ,T→ℝ,d0…

Q: Find a continuous function. f:D → R (where R is the real number line) such that f((x1, y1)) = f((x2,…

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Q: 3) Consider the topology 7 = {G CR: 0 ¢ G or G = R} on R. Is open T 3= interval (-1,1) compact in…

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Q: IS V is finite dimensiona! oad f is a homomor- Ex 6. phisni of V into itself which is not onto prove…

A: Given: V is finite dimensional and f is homomorphism of V into itself such that f is not onto. To…

Q: Let f be a holomorphic function on the disc DR centered at the origin and of radius Ro. (a) Prove…

A: Here we have given that f be a holomorphic function on the disc DR0 centered at the origin and of…

Q: Prove that f : R → R by f(x) = 2x+4. Prove that f is one-to-one and maps surjectively onto R.

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Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: Prove or disprove whether the function f (r) = a +2 is a bijection from R R %3D

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Q: Let A = Z+ x Z+. Define a relation R on A as follows: For all (x, y) and (z, w) in A, (r, y)R(z, w)…

A: We will prove the given statement.

Q: Let {Xa | a E A} be a collection of Hausdorff topological spaces. Prove that IαEA Xa is Hausdorff…

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Q: Sketch the following sets of points in the x-y plane. {(x, x^2): x is in R}

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Q: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

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Q: Let A = Z+ x Z+. Define a relation R on A as follows: For all (r, y) and (2, w) in A, (r, y)R(z, w)…

A: Given that: A=Z+xZ+. Relation is:        For all (x, y) and (z, w) in A, (x, y)R(z, w)⇔x+w=z+y…

Q: Let a, b, c, d be real numbers where c and d are not both zero. Let X = {x E R | cx + d + 0}. The…

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Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Q: A mapping f:C→ C is called bounded, if sup |f(2)| < o0. zEC Show that sin, cos : C →C are unbounded…

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Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

Q: Let R be the ring of continuous functions from R to R. Prove that M = {f e R : f(0) = 0} is a…

A: To show that the given subset M of R is a maximal ideal, we first show that M is a ideal of R and…

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

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Q: Consider E = C( [o,(]) as normed spaces with normS II f | = sup 1fcxl and l fll,=Sifwildx. %3D Show…

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