Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

Q: Q3 Iet X-{mingP;q } and ΦΧ 3. min is 2 asigma algebra over over X ?

A: Given that  X = {m, n, p, q} and ∑=ϕ, X, {m}, {n}, {p, q}, {m, n}, {n, p, q}, {m, p, q} we have to…

Q: let 9(1)- x 96)2+5x+ī in Zs[x]. Find in Z(x Such that where either (x)=0

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Q: O Show that Z4 (under addition) is isomorphic to Z% (under multiplication

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Q: Let H x K = { (h,k) |h in H, k in K}such %3D that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), %3D for…

A: the answer is 4 see the calculation

Q: Let a ≠ b in a field F. Show that x + a and x + b are relatively prime in F[x].

A: Definition of relatively prime: A polynomial in fx and gx in Fx is said to relatively prime if the…

Q: A O Type here to search

A: Figure A is the required answer.

Q: Let ficxT)→(メッテ)and g: メっテ)→(Kっズ) e are homomorphism then gof 1s homo.

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Q: List all the homomorphisms from D5 to Z2 + Z2.

A: There are 4 homomorphism from D5 to Z2+Z2

Q: 2. Let ACR. If a is a lower bound of A, then it follows that -a is an upper bound for A.

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Q: Suppose that {$1,...Øn} F p and I F ø; for all i < n. Show that I F þ.

A: Proof: Suppose ϕ∈Γ∗ϕ∈Γ∗. As Γ∗Γ∗ is maximally consistent it clearly follows that there exists a…

Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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Q: Let n be a positive integer. The coefficient of x°y° in the expansion of (1+ xy+ y²)' equals

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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: Let {v1, ..., Vp} be an orthonormal set in Rn. Verify the following inequality, called Bessel's…

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Q: Show that is isomorphic to Z2.

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Q: Let a, b € Z with a | b. Prove that an | bn for every positive integer n.

A: Given that           a, b∈ℤ with a|b. To prove an|bn  , we shall use mathematical induction.

Q: Q3. (а) Let X %3D {m, п, р, q}, and 3 {Ф, X, {m}, {n}, {p, }, {m, п}, {n, p, q) Is a sigma algebra…

A: It is given that

Q: Prove that d24 and s4 are not isomorphic. List elements and there order.

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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 the order of Z_2 x Z_2.

Q: Prove that all elements in S are multiples of 3. by Structualindrction 3ES X,Y ES > (X y )ES

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Q: Evaluate det(A) by using expansion along a row or column, given

A: Given:         A = 01201345-20-111-100   To find :     det(A) using expansion along row or column

Q: Define injections f : N → N and g : N → N by f(n) = 2n and g(n) = 2n. From f and g, the…

A: Given  f:N→N and g:N→N are two injective function defined by f(n)=2n and g(n)=2n Let 2n = k then…

Q: Evaluate det(A) by a cofactor expansion along a row or column of your choice. 5 0 0 1 4 4 4 -1 0 A =…

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Q: O Prove that the composition óf önlo

A: # we are entitled to solve one question at a time ,please resubmit the other question if you wish to…

Q: *** Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.***

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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 K be an extension of be an algebric element o 2 F Show that a ² is algebr

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Q: Let G = Z, O Z, H = {(0, 0), (2, 0), (0, 2), (2, 2)}, and K = ((1, 2)). Is GIH isomorphic to Z, or…

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Q: Showthat S subring of RIKRZ+R3

A: For the solution of the problem follow the next step.

Q: s] Assume that is an operation on S with identity element e, and that x * (y * 2) = (x * z2) * y for…

A: A operation  " o " on S is said to be commutative if x o y = y o x  for all x and y in S  And…

Q: p be an irreducible reprsentation of G

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Q: > a sigma algebra over X ?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Prove Theorem  A + (B + C ) = (A + B ) + C .

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Q: 2. Let f and g be permutations. Show that the fgf¯ is the same as the order of g. order of h =

A: our objective is to prove the result

Q: Which sequence of transformations maps ABCD onto A'B'C'D' and then maps A'B'C'D' onto A"B"C"D"?

A: Answer is option (1).

Q: What is Wronskian of e' and et? 1. -2 -2t O000

A: Given that  e^t and e^(-t)  To find the wronskian of e^t and e^-t

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: Evaluate det(A) by a cofactor expansion along a row or column of your choice. 3 0 0 1 0] 4 4 4 1 0

A: Given that

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

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Q: Let X be an infinite set. Define u : E = P(X) → Ř by %3D if A is finite µ(A) = 1 if A is infinite.…

A: Let X be an infinite set and ∑ =P(X) Let μ=P(X)=∑ → ℝ is given by μ(A)=0if A is finite1if A is…

Q: Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).

A: We have gcd(-a , b)=gcd(a,-b) let gcd(-a , b)=d Where -a=de  , b=df  ⇒a=d(-e) , -b=d(-f)⇒d is…

Q: Let A ∈ Mn×n(F). Under what conditions is det(−A) = det(A)?

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

Q: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

Q: Find infinitely many polynomials f(x) in Z3[x] such that f(a) = 0 forall a in Z3.

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Q: Let X be a rs from Exp(0). Construct a (1 - a) CI for 0.

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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.