Prove that G and H IS sube GroP form CX H

Q: on G by a * b = ba. Prove that G is isomorphic

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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: Prove by structural induction that F C F2

A: Given that, F1 such that, 3∈F1, n∈F1⇒n2∈F1. and F2 such that, 3∈F2, n,m∈F2⇒nm∈F2. Consider Pn be…

Q: In a Hilbert space. Prove that xn→ x provided ||x,|| →||x|| and (xn , x) → (x, x).

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Q: Prove that. E) define and give example

A: 1. Binary operation 2. Idempotent element 3. Odd permulation 4. Cyclic group

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: 5. Suppose that f: X →Y and g : Y → Z are bijections. Prove that gof is a bijection and that (go f)-…

A: We will find out the required solution.

Q: Prove that K = {x+yv3 ] x, y E Z} is a subring of R.

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Q: 2. Show that A is Hermitian if and only if (|A|) ER for all |t).

A: The given question is related with functional analysis. We have to show that…

Q: Let u be Find all t

A: Given utt=π2.uxx u(0,x)=cosx ut(0,x)=0 u(t,x)=0 for all x∈ℝ

Q: Prove that z =z iff z is real.

A: z represents a complex number which is in the form x+iy, where x and y are the real numbers and i is…

Q: ZW and ZX are supplementary and ZW and ZY are supplementary. Construct a proof that ZX = ZY.

A: &lt;W  and &lt; X  are supplementary angle  the sum of supplementary angles = 180 degree &lt;W  and…

Q: Prove that Z3 x Zz is isomorphic to Zg.

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Q: Let X e L'(N, F,P) and let G,H be sub o-algebras of F. Moreover be independent of o(o(X),G). Show…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Prove that f : R – {3} → R – {1}, defined by f(x) = is a bijection. x-1 x-3

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Q: Prove that c = cn. i=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: If X has full rank, so that P = X(X'X)-'X', prove that C(P) = C(X).

A: Given information: It is given that X is a full rank matrix.

Q: Let L be the operator on P3 defined byL (p(x)) = xp'(x) + p''(x) If p(x) = a0 + a1x + a2(1 + x2),…

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Q: Let E be the set of all positive even integers. Prove that E is countably infinite by defining a map…

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Q: Functional theory and functional analysis

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

Q: Prove that if H ≤ G, then aHa – 1 ≤ G, for all a in G.

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Q: 5. Let 2x g(x) = ,0 3 find (go g)(x)

A: Given function, We know that ( g o g )(x) = g(g(x) ).

Q: Let T={a: a ER}, then T is Borel algebra.

A: Borel sets of real numbers are definable as follows. Given some set, S, a σ-algebra over S is a…

Q: Suppose A=[v_1...v_n]. Prove that if A(x)=b is consistent if and only if b is an element of…

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

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Q: Prove that R* ⨁ R* is not isomorphic to C*.

A: First, we count the elements of finite order; R has two elements of finite order that is (1, -1) C…

Q: Let G=(V, E) be a finite

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Q: - Derive formulation for the elements of c(m, m') modified Daniell Kernel

A: Given information: The information about modified Daniell kernel is given.

Q: Let a in F_p, a is non zero then show

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

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Q: Prove that X ∈ ℘(A) if and only if X ⊆ A

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Q: Show that cl(a) = {a} if and only if a ∈ Z(G).

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

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Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: Prove that for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

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Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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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: 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: Let A be compact and U C A be closed. Prove that U is compact.

A: The complete proof is in given below

Q: Let P = {X E P(Z*)|X is finite}. Prove that P is denumerable.

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Q: Prove that where e, > (1,0)"

A: Prove that     conve1, e2, -e1, -e2=x∈ℝ2 : x1+x2≤1 where e1=1, 0T , e2=0, 1T.

Q: (d) Prove Ker(f) = 1/J (e) Apply the Fundamental Homomorphism Theorem (FHT)

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Q: Let x and y be coprime positive integers. Prove gcd(x + y, x² + y²) E {1, 2}.

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Q: Prove that for all k E N, k ose [coske -sink0j cose Lsinke cosko

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Q: 6. Find an explicit function f: [0,1] and prove why it is (0, 1) that is a → bijechove bijection

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Q: Let E be the set of all positive even integers. Prove that E is countably infinite by defining a map…

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.