Let h: (Z, +) (R – {0},.) define as h(n) = n² then h is 1-1 O 1-1 and onto onto No one O O O

Q: Let S be a non-empty open connected subset of R². The number of possible continuous onto functions…

A: Please see the attachment

Q: Suppose that j and k are positive integers such that j< k. Give a combinatorial proof of t identity:…

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Q: Let f: R R given by f(z, 9) = (x+2y, z- y). Of is 1-1 of is not 1-1 of is not onto of is onto

A: Given that fx,y=x+2y,x-y

Q: Prove by bijective (one to one onto) of function 2" f: 2ln] P(({0, 1}, n)).

A: In this question we have to deal with the cardinality of a function.

Q: Еx6. IS V is finite dimensional ond f is a homomor- phisni of V into itself which is not onto prove…

A: If f is a homomorphism of V into itself, then f(0) = 0 .

Q: Prove that det(At) = det(A) for any A ∈ M2× 2(F).

A: Let us consider a 2*2 matrix A whose entry belongs to F, which is given below

Q: Prove that for all integers n > 2, f(x) = x/n is continuous on [0, 0).

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Q: Let . be the operation on the cartersian product of real numbers R • R such that (a,x) * (b.y) =…

A: Given (a,x)*(b,y)=(ab,x+y)

Q: 4. Use mathematical induction to prove that Vn E Z* u {0}, []²=o(2i + 1)! > ((n + 1)!)**. n+1

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Q: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: 3. Use the above Lemma and Mathematical Induction to generalize the previous exercise. That is,…

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Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: The objective is to find Z_2×Z_2.

Q: 2. Let R = {(x, y) : x, y E R}. Prove or disprove that R is reflexive, symmetric and/or transitive…

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Q: 9. Let be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a…

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Q: f:A→Z is not onto and f:A M is onto.

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Q: 28. Prove by induction that i(i – 1) (n+1)n(n– 1) i=1 holds Vn E N,n 2 1.

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Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: Prove that for a E N, a always invertible in Z,2+3a+1·

A: We know that an element a∈ℤn is called invertible if there is an element b∈ℤn such that a+b≡0  mod…

Q: Functional theory and functional analysis

A: Consider the provided question, Let A be an invertible operator acting on space X.

Q: Let h : N x N → N be the function h(x, y) = 3x + 2y. Use strong induction to prove that if n EN and…

A: Given that h : ℕ×ℕ→ℕ be the function h(x,y)=3x+2y. Then we have to prove by using strong induction…

Q: (b) If P is the projection on ce M of a Hilbert space H th riant under an operator T

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Q: (b). Let Y = {a, e, o, i, u}, take two relations R and S on Y and Prove that (SoR)-1 : R-1os-1.

A: composition of relations

Q: 3. Let E be a bounded subset of R such that E = U₁1 Ej. Prove that if the E, are of Lebesgue measure…

A: Lebesgue Measure : A set E⊂ℝ is said to be Lebesgue measurable if for every subset of A of ℝ…

Q: Let Z ~ N(0,1). Prove for all z 2 0, P(Z > x) S

A: It is an important part of statistics. It is widely used.

Q: Prove that det(At) = det(A) for any A ϵ M2x2(F).

A: Suppose we are given a square matrix A with four-element: a,b,c,d A=abcdAt=acbd

Q: Let A E Mnxn (F) and B = aA, with a E F. Prove by induction that det(B) all positive integer n. a"…

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Q: Let H(x):=k for x=, (keN) and H(x):=0 elsewhere on [0,1] Show that H(x) 4ER[0,1].

A: In this question, we use the definition of Riemann integral to check the function H(x) is Riemann…

Q: Let T be the linear thansformation from R5 to Rª defined by X1 1] X2 X1 1 3 2 X2 -8 1 -2 2 0 -1 4 X3…

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Q: Let H = {0, ±3, ±6, ±9,…}. How many distinct left cosets of H in ℤ are there?

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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: In the domain of Gaussian integers Z[i], the element 15 is 青 O Reducible O Unit O Irreducible O None…

A: Since 3∈ Z[i] and 5∈ Z[i] And 15 = 3 × 5 Hence 15 is reducible element in Z[i]

Q: Let (X,T) be a topological space,  K a compact subset of X and (Fn)n∈ N a family in P(X) with Fn≠ø…

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Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: 3. Find the generator of X = Xo + ut + oWt, where W is a standard BM, and identify the PDE satisfied…

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Q: Suppose  there exist transposition f_i , 1<=I<=t  such that

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Q: Let D be an integral domain with four elements, D = {O, e, a, b }, where e is the unity.   Prove…

A: Let D be an integral domain with four elements, D=o,e,a,b, where e is the unity. The objective is to…

Q: 7. Prove there exists a unique x E R such that for every y € R, ту + т — 2 3D 2у.

A: We have to find the unique value of x for which the given equation is valid. We have, xy + x -2= 2y…

Q: Let z = x + yi, z2 = 1– 0.5i , and z, + z2 = 3 – 1.5i. Find In(z,) – In(z2) %3D A) In(25) ± 2niN D)…

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Q: Suppose that f: C C is a holomorphic function such that there is a fixed kEN for which =1, for all.n…

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Q: Is the function f(n) = n - 5 from Z to Z (where Z is the set of integers) (a) one-to-one? Explain…

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Q: i. Prove that gi : Z → Z; g1 (x) = x – 4 is both one-to-one and onto. ii. Prove that g2 : R → R;…

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Q: 1) Let F := {B E B(R) | B = –B}.Show that B is a o-field.

A: This is a question from Measure Theory.

Q: Let o,T E Sn for some positive integer n satisfy o(k) 1< k < n. Prove that o-(k) = k and p(k) = k…

A: Solution: Let us consider the symmetric group Sn, for some positive intiger n.  We know that Sn is…

Q: If V is finite dimensional ond f is a homomorphisni of V into itself which is not onto prove that…

A: If f is a homomorphism of V into itself, then f(0) = 0 .

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

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Q: Which of the following form an orthonormal set in a euclidean inner product space R 3 ? 101 A. 1. 0,…

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Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below

Q: Prove that det(kA) = kn det(A) for any A ∈ Mn×n(F).

A: In determinant the scalar quantity is taken common from a single row or a column.