4. Prove: Let a, b e Z*. If a|b and bla, then a = b or a = -b.

Q: 2 Let A | and U = H. Then A2017U is equal to = ... A) B

A:

Q: Let E = Q(V3, i). Then, Gal(E/Q) is cyclic. %3D False True O O

A: Given  E= Q(3^1/2,i) Gal(E/Q)  is cyclic .

Q: Consider the Venn diagram below. Illustrate the other De Morgan's law by selecting the region(s)…

A: GivenAc ∪ Bc

Q: at II Prove that if a;-ta and b;→ aj + q sa+b. →a and b.

A: See the attachment

Q: Let A and U = H. Then A2017U is equal to

A: The given matrix is A=-120-1 find A2=-120-1-120-1=1+0-2-201=1-401 therefore A2=1-401 similarly…

Q: If A= then A" + nl is equal to

A:

Q: Let (W, t > 0) be standard BM. Show that P(n|w,1 – Wa sc) → as n → 0.

A: From the given information,  (Wt, t≥0) be standard Brownian Motion. Note: In the given situation,…

Q: prove that: DO e de a e +6 2/3 cwhere a, b > o

A: Where a, b>0

Q: Let X ∼ Bin(n, p), and let Y = n − X. Show that Y ∼ Bin(n, 1 − p).

A: Here X follows Binomial with parameters n and p. Here, Y=n-X. Then Y takes the values n-0, n-1,…

Q: If E and E, are measurable , then (E,UE,) is not measurable. ylgn

A: False.

Q: Let w = z. In(xy?z³), then which of the following is :? O a. O b. c. O d. 0 Oe.

A:

Q: d set): Let (M ,d) be a (SUT)' = S' U T'.

A:

Q: 4. If X and Y are independent RVs, then O2+ ,(@) = ¢,(@) × ¢,(@)

A:

Q: find the'mters ection d Exe sy,z]= [a,0,31++ 4,-2,178 [+sy,z]=[6,-12,--11]+5 [1,2,4]?

A:

Q: Let A,BE Mmna Shaw that vTAw= V' Bw for all VER™ end WEIR" if and only if LA=B. %3D

A:

Q: 6. Let a, b e Rwith b + 0. Show that R(a + bi) = C. %3D

A: We have to show that R(a+bi)=C. We prove it in two oashow that R(a+bi) is a subset of C and C is a…

Q: locally Lipschitz in (t,x)E[a, b]x D.

A:

Q: Suppose A C B, a E A, and a ¢ B – C. Prove that a E C.

A:

Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

A:

Q: Let c E R and suppose that an → a. Show that can → ca.

A:

Q: for AUB unih AG B inderser for is not unusual.) Then, A+B= B+ A A B= B A A (B+ C) = A B+ A C A+Ø= A…

A: In this question we need to explain about the underlined symbols ∪ ,   ∩  ,   ϕ  .

Q: Show that if A is nonsingular and AB = O for an n × nmatrix B , then B = O .

A: We have given  matrix A is non singular and AB=0 for an n×n matrix  B. We have to prove that B=O

Q: Suppose, X0- Binom(m,0)and 0~ Beta(a, B), then E(X) is equals to, a) maß b) aß ma c) d) me

A: Given: Suppose,  Xθ~Binom(m,θ) and θ~Beta(α,β)

Q: (a) Prove that de 2n a>b>0 Jo a+ bcose Ja? - b2

A: We will use basic knowledge of Integral Calculus and trigonometric identities to solve this…

Q: Let V = {cities of Metro Manila} and E = {(x, y) |x and y are adjacent cities in Metro Manila}. %3!

A: Given: Let v= cities of Metro Manila  and E=x,y/ x and y adjacent cities in Metro Manila The…

Q: (6) If w = f(u, v), where u = x + y and v = x – y, show that = 2 ди

A:

Q: (a, o) is not measurable. ylgn ihi

A: Given :-(a, ∞) is not measurable.

Q: When is S \ T = T \ S for sets S, T?

A: Given that two sets S and T. The definition of S\T=a∈S,a∉T and T\S=b∈T,b∉S. Here, neither of these…

Q: SHOW that V Ź a? Jor a, E IR when His ineguaity e kold .

A:

Q: Q Let 8=R and Prove disprove that zis atopdagy or on R.

A:

Q: Let a e Z. Prove that if 7 | 2a, then 7 | a.

A:

Q: Let SCR. If sup S (resp., inf S) exists, then it is unique. ylgn ihi

A: The supremum (or least upper bound) of a set S ⊆ R which is bounded above is an upper bound b ∈ R of…

Q: If (a + bx)e = x, prove that x’y, =(xy,-y)

A: Since you have asked multiple questions , we will solve first question for you. If you want any…

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: 2. Let A € M, (R) be given. (a) If AA = 0, prove that A = 0₁. Does this hold if A € M₂, (C)? (b)…

A:

Q: Suppose A · B = 0. Show that this implies that C(B) is contained in Null(A).

A: Suppose A.B = 0 then we have to show that  Column space of B contained in the null space of A  i.e…

Q: 9. Prove that the following pair of formulas are equivalent in SDE. U → -(E V D), (-E A ¬D) →¬U

A:

Q: 4.2. Let a, b E Z, where a 0 and b 0. Prove that if alb and bla, then a = b or a = -b. -b.

A:

Q: c If in is an involution of X, show that (r, y)" = (yi", xi") is an involution of X2). %3D

A: Given X is a lattice. Given in  is an involution of X To show: ((x,y)in=(yin,xin) is an involution…

Q: 2) Let mf lu and (a,m)=| a,bE Z such that Let h= Om ( a) show and and (h,m)=1, %3D let. ne OmCab)…

A:

Q: Let X1,., X, N(u, o?). iid N(H, o?). Find E(), if ju = X1 + (X2 + + X(n %3D 2(n-3) 17 16 4 O 17(n-2)…

A: Given that, E(X1)= E(X2)= ......E(Xn)= mu

Q: (b) Using Venn Diagram to prove that (A U B)e = A° N B° ?

A: To prove the following using Venn diagram A∪Bc=Ac∩Bc First consider the left hand side  Set A

Q: 2. Let A € M, (R) be given. (a) If AA = 0n, prove that A = 0₁. Does this hold if A € M₂, (C)? (b)…

A:

Q: show that fo=Ixl is Conlindalus everyuhere

A: Since you have posted multiple questions, as per our policy we will answer first question. Please…

Q: 5. Let TE B(H). Show that (ker T)- = ran T.

A: Given: T∈BH and T† is an adjoint of an operator T. To show: ker T⊥=ranT†

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: 4. Let a,b,c E Z where c| (a+b). Prove that gcd(a,c) = gcd(b,c).

A:

Q: 3.1 Prove: For all a and B in R, Ja + B] > | |a| – |B||. Hint: la + B] = |a – (-B)|-

A: Answer is in below steps.

Q: 2. If (x, y) is a sa

A: As per the guidelines, we are supposed to solve one question. Kindly repost the other question as a…

Q: 4) Let x, y, z E Zg. Prove or disprove the following: Ifxy = xz, then y = z.

A: