Let C = {x ∈ P(A) : ∃y ∈ x, y ∈ A and y /∈ B}. Prove that P(A) − P(B) = C.

Q: 3. For f(x, y, z) where x = u – v, y = v – w, and z = w – u: - Compute - Show that of - af af and af…

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Q: Given f(x,y, z) = In(x² + y² + z²), show that ² f , ²f , ² f x² + y2 + z2

A: Given problem:-

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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Q: Prove that if w=f(x, y, z), x=r-s, y=s-t, and z=t-r then dw dw dw + + = 0. dr ds ôt

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Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

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

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Q: let H = {n² |n E z*}, determine whether it is closed under

A: Given : H={n2|n∈Z+}   To Check:  H is closed under binary addition or not

Q: let H = {n² |n e Z*}, determine whether it is closed under

A: Here, we have H=n2| n∈ℤ+. To show the set is closed under binary multiplication. This means we have…

Q: (b) Prove that xe" 2 for some x in (0, 1). %3D

A: By the intermediate value property of continuous function if a function f:(a, b) --> R be a…

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

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

Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

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Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

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Q: The domain of the function f (x, y, z) = In(6 – 2æy) is O {(z, y) E R² : zy > 3} O {(z, y, 2) E R° :…

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Q: Prove that if Y:nz+P(x- mz) So thdz

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Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

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Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

Q: Given f(x, y, z) = xV/x² - yz - - Determine of дz . O of дz of дz af дz of дz = = x² - у 2√x² - yz…

A: Find the derivative

Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

A: Consider the given inequality 0≤x2-xy+y2  for all x and y in  ℤ Now,…

Q: Suppose that C> 1. Prove that if t > Vm ln C+In C, then ()/F) >C.

A: We will use the following Strining's approximation :  n! ≈ √(2πn) e-n nn To prove the given…

Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: Where x = (X1, X2) E R². 3- N: R2 - → R defined by N(x) = /x1² + x2² is a norm on R². (x1, x2) E R².…

A: Let X=(x1,X2)€R^2

Q: Let r, y EN such that y # 0, then y+r#0.

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Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

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Q: 2. Compute P, and Py as a function of x, y, P.

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Q: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

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Q: Let f(x, y, z) = x²y°z – xyz², and compute: (a)

A: In this question we will use basic of vector  calculus i.e gradient of any function and curl of…

Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

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Q: if x, y>o such that x Sy then Vã svy Prove by direct method

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Q: Q1:- Let T =- where W-N(0, 1) and V~xShow that T2-F(1,r).

A: We know that  The ratio of chi-square divided by its degrees of freedom is called the…

Q: ; Let x, y ER" be given. (a) lf ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

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Q: Let x, y ∈ R. Suppose x2 + y2 = 0. Prove that x = 0 and y = 0.

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Q: Let E = XE[0, + 0) } then Sup E Маx E- %3D %3D x+ O True False

A: In this problem we have to determine Sup E and Max E, and check Sup E =Max E is true or false.

Q: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

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: Show that for any positive integers i and j with i > j, we have T;(x)T;(x) = [Ti+;(x) +T;-j(x)].

A: we have to show, for any positive integers i and j with i&gt;j TixTjx=12Ti+jx+Ti-jx Now, we know…

Q: If Z = f(x + g(y)) , prove: Zx * Zyx = Zy * Zxx

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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: If Rolle;s theorem applies on f(x) = 9 – (x – 3)², there exists a 'c' in [0, 6], Which satisfies the…

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Q: əz Let z = x² + y³, x = 2st and y = s-t². Compute for and ar

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Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: Prove (x + y) + z = (x + z) + y without using induction.

A: It can be solve using group theory

Q: Let x(v, w) = } In(v² + w²). Show that = 0

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Q: Let f(x) = 2x + 1 in Z4 [x]. Then f(x) has a zero Z4 as a subring.

A: Let f(x) =2x+1 in Z4x. Then f(x) has a zero in any ring containing Z4 is a subring_____?____

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: Prove that if w = f(x,y,z), x=r-s, y=s-t, and z =t-r then ôw ôw ôw + âr' as = 0. ôt

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

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Q: Prove that if X ⊆ Y then X ∩ Z ⊆ Y ∩ Z.

A: To prove this we will take an element from X∩Z and show that 'a' also belongs to Y∩Z.