Q Let ) be a o- Algebra on X. Assume that ) → [0, 00) are two measures on X.Then a. 3y +2µ is not a measure on X. b. 5y+ µ is a measure on X. c. 2y- 3µ is a measure on X. d. y-u is a measure on X.

Q: (6) Let (X,M.) be a measure space, and {Ej}cM Prove that : (I) If E,SE,S…. ,then j (II) If E,2E2...…

A: The given problem is related with measure theory. Given that X , M , μ is a measure space and Ej1∞ ⊆…

Q: Given the bases B = {u,,u2} and B'={u'1, u'2} for R², where and u'1= The transition mat rix Pg-8'…

A: Given bases  are B=u1,u2 and B'=u1',u2'  where u1=10,u2=01 and u1'=12,u2'=11 To find the transition…

Q: Let J={a: aER}, then I is Borel algebra.

A: False Statement' Detailed Explanation

Q: det (NT) is sPace Toflogecu! and T=0U{un}such that 7Show that (N, Cl is compact slase

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Q: 5. Let (, ) be the Euclidean inner product on R", i.e. (x, y) = y"x %3D for r, y E R", and let || -…

A: In the given question question given that  inner product on Rn is defined as <x,y> = yTx and…

Q: Q4) Consider the topology Y= {ACR:OGAJufo} on R ad A- RIF0}. Prove that CA is the discrete topology…

A: Given ζ=A⊂ℝ:0∈A∪∅ is a topology on ℝ. To show ζA is the discrete topology on A. By definition,…

Q: 3. AABC is isosceles with AB = AC; M and N are the midpoints of AB and AC respectively. If 41 = %3D…

A: Use the congruency property

Q: Let X =[0,2).Define r= {[0, a):0sa s2). 1. Show that r is a topology on X. 2. Give an example which…

A: Let X be any set and ζ be the collection of subsets of x then ζ is called as topology if it contains…

Q: Which of the following is a topology on R? {UCR: U is infinite}U{U<R: U is countable} O {[a, b): a,…

A: We will use the definition of Topology.

Q: Given the bases B={u , u2} and B'={u'1, u'z} for R2, where , U2 and u'2 = The transition mat rix…

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Q: Let R be the set of real num bers and let {C: ]: ae R\ {o} a be R}. : a € R\ {0} ^ be O Show that…

A: Use the definition of group action

Q: Which of the following is not a base for any topology on R? a. {(a, b]: a,b ER and a <b} O b. {[a,…

A: Base for a topology: The base for topology of any set X is collection B of subsets of X which…

Q: Let T E L(U, V) and S e L(V, W). a) Show that if S is injective then nl(ST) = nl(T) and rk(ST) =…

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Q: 1. In R" where ne N, let B,(x) = {y: |x – yll 0) such that B,(x) C E} Show using the definition of…

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Q: 2) What rule maps ABC onto A'B'C'? A. (x,y) → (x+4, y = 4) B. (x,y) → (x-3, y + 5) 4 (2-0)-²00) AY 5…

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Q: Let X = R, and d be the diserete metrie, and A= {1,2) Show that if A is compact or not

A: Given set A={1,2}

Q: In Euclidean metric space (R, I. I), prove that (1+1)→ 1 as n → ∞o in R.

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: Qt Let T= {carb) a,b ER\fopp and of defined onI by Carb)*(c,d)= (ac, bd) for all (a,b), (c,d ET,…

A: The set T is defined as, T=a, b | a, b∈ℝ\ 0 The operation * is defined on T as follows, a, b*c,…

Q: Which of the following is not a base for any topology on R? O {(-0,a): a E (-∞, 0]}U{(0, ∞)} O {(a,…

A: Base for a topology: Let X, τ be a topological space. A collection of open sets B is said to be base…

Q: et S be a nonempty subset of R and let k E R. Define kS {ks : s E S}. Prove if k > 0, en sup(kS) = k…

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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&lt;x2 where ℝ2 and ℝ are…

Q: Which of the following sets has oo in its interior: (a) {z = « + iy, x² 0} (c) {z = x + yi, |z +1|…

A: We have to find region bounded by given in equation and find which region goes to infinity  Here we…

Q: Given the bases B= {u,,u2} and B'= {u'1, u'2} for R, where U2 and The transition mat rix Pg-g from B…

A: We first find the coordinate vectors of u_1 and u_2 relative to basis B' ,then we will form the…

Q: Let T={[a,0) : aER}, then T is not Borel algebra

A: Given a statement

Q: Prove: If U is a neighborhood of x0 and U ⊂ V, then V is a neighborhood of x0. Also, if U1 ..., Un…

A: The neighborhood of a point: Let (X, T) be a topological space and x0∈X. A neighborhood of x0 at is…

Q: Recall that for D a squarefree integer, the norm function N on Q[VD] is defined by N(a + bVD) = (a +…

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Q: If z, w E C, prove that z+w=z+w.

A: This follows from simple substitution. The detailed solution is presented in step 2. As requested i…

Q: Given the bases B = {u,u2} and B'= {u', u'2} for R?, where and u'= The transition mat rix Pg-g' from…

A: Linear transformation problem.....

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: Let 1={Ø}U{UER:NC U}, -n+3 ). A = :n EN and B = [0, ∞). Then n+1 in the topological space (X, 1) O…

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Q: Let T={(a, 0): aER}, then T is Borel algebra.

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

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Q: Decide if the statement is True or False. Justify your ans why not, and give an…

A: a) The poset (P(Z), ⊆)

Q: Let T={(a, o): aER}, then T is Borel algebra

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Q: Then ||-|| is a norm on R.

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Q: Q1/ Prove that (Sa,0) is not Comm Sroup.

A: To prove the non commutativeness of the group

Q: a) Suppose that you wish to prove that ARST = ASRV. Using the reason Identity, name one pair of…

A: Given: We wish to have the triangle RST and triangle RSV to be congruent.

Q: Let J={(a, ∞ ) : a€R}, then T is Borel algebra. ulgn O

A: Given a statement

Q: Let R be a UFD and fx) in REX) So, 0+a € IR.. Show that claf)~ ac(f) a nonzero polynamial

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Q: Let T={a: aER}, then T is Borel algebra.

A: Given a statement

Q: let E be a set that baunded. Prove that Inf E = Sup -E) where R, XE E y

A: To prove for a buonded set E inf(E) = -sup(-E) , where -E = {-x, x belongs to E} , we first prove…

Q: In the semigroup with identity (R,) we define Va E R+, fa: (R, +) → (R, +) by Vx € R, fa (x) = ax.…

A: The given map is fa:ℝ,+→(ℝ,+), defined as fa(x)=ax, ∀x∈ℝ. We have to check the following: (a). Is…

Q: prove that for operatrs a and b which comute with thier commuters[a,b]=c,so that [a,c}=[b,c}=0 ,…

A: We know that ekABe-kA=B+kA,B+k22!A,A,B+... Given that A and B commute with their commutator then…

Q: Let A EMn (R) be given. If 11x11 = 11yll = 1 and x •y = 1, show that x=y

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Q: Let J={(a,∞): a€R}, then T is Borel algebra. صواب İhi

A: True

Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

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

A: See the attachment

Q: be such that Ca,m)=1, Let thE N and a,be Z Cb ,m) = ), and Cab, m)=\. Let j= Om Ca); k= Om(b) and n=…

A: Let m,n∈ℕ and a,b∈ℤ such that a,m=1, b,m=1 and ab,m=1 Suppose j=Oma, k=Omb and n=Omab Again, let…

Q: 1. Let x, y, z€ R, w e R*. Also let a, b, ceZ and de Z*. For each of the following determine is…

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Q: Consider the standard topology on R, Let A=[3,7], the interior of A is Oa A O b.6 Oc. (3,7) OdR

A: Consider the standard Topology on R. Given A =[3,7] We need to find the interior of A.