Let D be a Euclidean domain with measure d. Show that if a and bare associates in D, then d(a) =d(b).

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: If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

A: We will prove the given statement.

Q: In Question,determine whether the definition gives a norm.

A: p(x)=p(0)+p(1)-p(0) for p(x) in P1

Q: let H = {n² |n e Z*}, determine whether it is closed under

A: Here, we have H=n2| n∈ℤ+. To show the set is closed under binary multiplication. This means we have…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

A: We have, A=set of positive numbers  A is bounded above set so, for all  x∈A ∃ K∈ℝ such that x≤K We…

Q: Prove (t, to) = A(t)Þ(t, t,)

A: The variables of function t and to are independent variables.

Q: 25 If A E M, is symmetric, prove that AT is symmetric.

A: Symmetric matrix

Q: S. Use the theorems and given information to show that r

A: The figure is as shown below : Given, ∠4 = (13x-4)o, ∠8 = (9x+16)o and x=5

Q: Prove that on average, permutations of length n have H, cycles, where Hn i=1

A:

Q: Let D be a Euclidean domain with measure d. Prove that u is a unitin D if and only if d(u) = d(1).

A:

Q: defined BCX,x). a norm Dn

A:

Q: 1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B → R by…

A: we have to solve given problem:

Q: be any nonempty set and let µ* be an outer measure on P(X) and let A, B e P(X). If at ne of A, B is…

A:

Q: Let A, Be Mnxn (F). (a) Prove that if A~ B, then At ~ Bt.

A: From the given information. The matrices A and B belong to M of order nxn. It is required to prove .

Q: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

A: In the above question it is given that B is the set of all bounded sequences of real numbers and a…

Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

A:

Q: Find all values of m such that a(r) = {X – 1 if x m is continuous everywhere.

A: Questions from real analysis.

Q: Given a Euclidean space E, if f: E→E is any self fanguidodge door of pas food of go

A:

Q: Prove that on average, permutations of length n have H, cycles, where Σ Hn i=1

A: We will take an arbitrary set containing n elements and observe the required permutations.

Q: If E, and E2 are measurable , then (E,UE,) is not measurable . صواب İhi

A: Here we have given E1 and E2 are measurable sets

Q: lar, let m = 2 and n e of A and B such that

A:

Q: Let v e R" and letk e R. Prove that S = {x € R" :x•v = k} is an affine subset of R". %3D

A:

Q: In the Triangle Inequality of Euclidean n-space: Prove ||x - z || <= ||x - y|| + ||y - z ||…

A: The triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle…

Q: Let T={[a, ) : a€R}, then T is not Borel algebra.

A:

Q: . Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

A:

Q: 8. If (X, M, ) is a measure space and {E₁} CM, then (lim inf E₁) lim inf μ(E). Also, μ(lim sup E₁) ≥…

A: Given that if X, Μ, μ  is a measure space and Ej1∞⊂M .

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 +…

A:

Q: F(p), then prove that

A: simple use of integral transformation

Q: Let E and F be measurable sets each having finite measure. Prove that m(EAF) = 0 if and only if m(E)…

A: To prove: The statement "Let E and F be measurable sets each having finite measure. Then, mE∆F=0 if…

Q: Show that K, is not planar.

A: To prove: K5 is not a planar graph. Proof:  This is proved by the method of contradiction. We use…

Q: (a) Prove the following property: (:) - (.".)

A: Hey, since there are more than two questions posted, we will answer the first question. If you want…

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

A:

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

A:

Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: Let (X, E) be a measurable space and E e E. If E1 e £ is such that E1 C E and ƒ is measurable on E…

A:

Q: Consider R? and R3 with the Euclidean (c) norms. Compute || T || for T : R2 →R3 defined by T (x1,…

A:

Q: let Z be aranelom vanimble with Pelf (Z~Nco) Show that , the psf of Z² (Y=Z) is. chi-square

A: # Given : z be the standard normal variate  let y=z^2  then to show that y follow chisquare…

Q: Show that if A and B are similar, then AT and BT are similar.

A: Let A and B are similar matrices. By definition of the similar matrices.  The matric A and B are…

Q: Suppose f" exists and is bounded on Co,0), Prove that if bam fa)=0, then bim fGx)=0.

A:

Q: Let K = Span{ t-t, 1+3t}, then dim K = %3D

A:

Q: Let F1 and F2 be 4-dimensional flats in R6, and suppose that F, N F, # Ø. What are the possible…

A:

Q: (a) Prove that x, y = -1, y = 1 1 (b) Prove tha and n: m and

A: This is a problem of Number Theory.

Q: Use th properties of linear Space A upon t to prove s a) 0.X - o for all X EX b) a.x=0 =>(a=07v…

A:

Q: 8. If (X, M, ) is a measure space and {E₁} CM, then (lim inf E₁) ≤ lim inf μ(E). Also, μ(lim sup E₁)…

A:

Q: Let A and P be an n x n matricies, where P is invertible. Show that det(PA(P-1))=detA.

A: To show that,

Q: Let A = {x E Z* : x is a mutiple of 2 and x < 15}. Rewrite A in roster notatlon.

A: Given that : A = {x∈Z+ : x is a multiple of 2 and x &lt;15 }   Roster form : A representation of a…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.