a b 2.3 Let e: (M,(R), +) → be defined by e a + d. where a,b,c,d ER (a) Prove that e is a homomorphism. (b) Determine Ker e.

Q: 3 Let T : P3 R° be defined by p(0) p(1) Lp(2) T(p(x)) A. What is the nullity of T? B. What is the…

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Q: T: M2x2 (R) → R2 [x] is given by r(; ;) a С 3 (а +ь+с+d) + (а+ 2b + 3с + 4d)ӕ + (а + 36 + 5с + 7d)…

A: To solve this problem, we use the property  of linear transformation and to check condition which is…

Q: C). Define the map f: ZxZ → 2 by f(a, b) = a₂ (3) Is f one-one? (2) Find ker(f). Show that f is a…

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Q: 3. Let SP(la, b, c,d.e) and define W:SXSS as the union function: (A, B)AUB. for ill (A. B) ESXS. (a)…

A: Given: S=P({a, b, c, d, e})Define A:S×S→Ssuch thatA(A, B)=A∪Bfor all (A, B)∈S×S We see that S is a…

Q: The set G = :a is closed under addition %3D 0 2a

A: Use the closure property of addition.

Q: 6. Let f : R? → R defined by f((r1, r2)) = 11 – #2. Then determine if is (a) Lipschitz continuous(if…

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Q: the book is "First Course on Fuzzy Theory and Applications"

A: To  write down the table for the fuzzy composite relation

Q: Q3: Prove that X, = X, = ,X; = y %3D %3D %3D are symmetries of ys y"=3y" y"2+y''3 %3|

A: Given: To prove that the  X1 = ∂∂y,   X2 = ∂∂x,  X3 = y ∂∂ x are symmetries of  y' 5 y''' = 3 y' 4…

Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

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Q: J1 Q2. (a) Prove that ifE, and >, then nE, is a sigma algebra over X. (b) Is ,u, a sigma algebra…

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Q: 1 0 10 1 0 1 0 1 0 Let A = {1,2,3,4,5} and M, = 1 1 1 1 1 0 1 0 1 0 |1 0 1 0 1 a) Give the relation…

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Q: 6. Give an example of a relation R on A = {-1, –3, –5} such that: (a) R is both symmetric and…

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Q: q) Given f: Z1z Z12 , where f(x) =44 xE そ then f is a homomorphism OTruell O Fulse

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Q: Please fill in the blanks for questions 4.1.4A, and 4.1.4B.

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Q: show that L^1(R) a Banach algebra Commutative

A: We will define the multiplicative operation of Banach algebra in case of L1(R) and prove its…

Q: e Let A= 3 = 0. %3D ez ez o Show that A0. use mahix algebra to compute the product (エーA)(エ+A+A*).

A: We firstly find A3 and then compute (I-A)(I+A+A2).

Q: then L is a linear operator. Let L(x). R R be a mapping such that L( X2 Select one: O True O False

A: We will check whether the given statement is true or false.

Q: Let A E R"Xn. Show that =, Vx, y E R" iff A = A'. Show this inner product equality holds for A E Cxn…

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Q: a.) Ker(p) and p(24) for p : Z → Z7 such that Ф(1) %3D 4 b.) Ker(p) and p(16) for p : Z → Z10 such…

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Q: 1. (i) Let AC R. The characteristic function of the set A is defined to be Xa(x) = 1 if x € A and…

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Q: Consider X = [1,2]? with the relation (x1, y1)~(x2,y2) if x1y2 = X2y1. Also define g: X [1,2] by the…

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Q: Given n e N and the function f : R" → R", we can define, for each nonnegative integer k, the kth…

A: (f) Given : f : ℝn → ℝn is linear function,  where n∈ℕ. To show :  We have to check whether the…

Q: 12. Let N = {z e C* : - < arg z < }. Find a biholomorphism f : 2→ B(0, 1). %3D

A: Given- Let Ω = z ∈ ℂ× : -π4 &lt; arg z &lt; π4. To find- Find a biholomorphism f : Ω → B0, 1.

Q: 3. Show that the map T: R3 →R given by T(x, y, z) = (x+ y+ z, y – 2) is a linear map. Find the…

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Q: 0:C' →C',o(x)=x*an "ANYWAY. O SHOW THAT THERE IS A HOmomoRPHISm AND KER(o) DETeRmine %3D

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Q: ´f(1) f(2) T: R3 [æ] → R* is given byT(f) = f(3) . Which statement about T is correct? f(4). O Tis…

A: Isomorphism T or F

Q: Consider X = [1,2]? with the relation (x1, y1)~(x2, Y2) if x1Y2 = x2y1. Also define 3x g: X → [1,2]…

A: We shall solve first three subparts only as per company's guidelines. For others please post again…

Q: 9. Let F : R" → R". Prove that for each a the mapping h→ dF[a; h] is linear on R".

A: Given: F:ℝn→ℝm   To prove: For each a, the mapping h↦dFa;h is linear on ℝn

Q: 2. Let f : R × N → R be defined by f(x, y) = 2|x|y+ 1. Find the image of R×N under f, that is,…

A: This is a problem of set and function.

Q: 2. Which of the following maps is an isomorphism/ a monomorphism/ an epimorphism: (a). : (Sn, 0)→…

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Q: Q No 1. Let f: M → N be an R-homomorphism. Then if A is submodule of N, the f (f -1 (A)) = A N…

A: We will use basic knowledge of module theory to answer these parts. Note that this is a multipart…

Q: Q1/(A): Let f: M → N be R-homomorphism . Prove that : 1- f(0) = 0. 2- f(-a) = -f(a). Va EM

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Q: Which of the following is isomorphism? * f:(Z, +) → (R*, .) where f(x) = 2* O this option f: (Z, +)…

A: A mapping is said to be isomorphism if it is one-one, onto and homomorphism. A mapping f: X→Y is…

Q: a b Let e: (M2(R), + >→ be defined by e c d = a + d• where a,b,c,d €R (a) Prove that e is a…

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Q: Let (S,F,u) be a mee e B₁, B2 E F satisfying (E 1) = μ(B₂) for any pair of B₁) = 0. Then ao-algebra…

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Q: 1 11 1 0 1. Let A = (1, 2, 3, 4} and MR = Find the relation R defined on A. 0 0 L1 1 1

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Q: Consider X = [1,2]? with the relation (x1, y1)~(x2, Y2) if x1Y2 = x2y1. Also define 3x g: X → [1,2]…

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Q: Find det (T) for T:R- R, where T(x,,×2,*3) = (x,- x2, X2 - X3, X3-x).

A: We will find the matrix [T] from the image of the standard basis vectors of R³ under the…

Q: 6. Let f: R? + R defined by f(r1, 12)) = #1 – a2. Then determine if is (a) Lipschitz continuous(if…

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Q: Vhich of the following is isomorphism? f: (R, .) → (R, .) where f(x) = Inx O this option f: (R, +) →…

A: The map ϕ:A→B is said to be an isomorphism if: 1. ϕ is a homomorphism 2. ϕ is a bijection. A map…

Q: 23)The number of homeomorphisms from (X,Tinc,3) to (X,Texc.3) ( Where X= {1,2,3,4}) is : A) 4 B) 3…

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Q: Which of the following is isomorphisma f: (R, .) → (R, .) where f(x) = Inx this option O None of…

A: Lnx is not a function and second ome is not onto

Q: N J 9. Consider the permutations f and g = = 5 2 1 3 4 set {1, 2, 3, 4, 5). Then go f = 1 2 3 4 5 1…

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Q: Show that linear mapping  T : P2 (R) → R3defined by  T (ax2 + bx + c) = (a + b, a + c, b+ c), is an…

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Q: 7. Suppose that X and Y are disjoint sets and k E Z»o. Prove that the function p:U (P:(X) × Pk-:(Y))…

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Q: 1. (i) Let AC R. The characteristic function of the set A is defined to be xa(x) = 1 if æ E A and…

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Q: O: Find the general kolution ay ofte 1SE 1Order ODE of x²dY= Y- xy de

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Q: Q Let be a o- Algebra on X. Assume that y, µ :2– [0, 00) are two measures on X.T a. 3y-2u is not a…

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Q: 2. Let f : V → W be a homomorphism. Show that if the set S = {v1, 02, ... , Un} C V is linearly…

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