Prove that ∀y ∈ Σ*, P(y) is true, where P(y) is the statement "∀y ∈ Σ*, r(xy) = r(y)r(x)".

Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

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: a. Show that Ø, R E T. (by definition of Topology)

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Q: If the Wronskian of f and g is t cos t - sin t, and if u = f + 3g, v= fg, find the Wronskian of u…

A: It is provided that the Wronskian of f and g is tcost-sint. We need to determine the Wronskian of u,…

Q: Let x, y ∈ R. Suppose x^2 +y^2 = 0. Prove that x = 0 and y = 0.

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Q: 3;) Find the välie of k isó that. the furnictioin %3D .kx*. .if. x 2 2. is. continuous .cvery iw…

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Q: If la,} and {b„} are divergent then {a, + b,} is divergent. True False

A: Given: an and bn are divergent . The objective is to state whether an+bn is divergent is true or…

Q: 5. prove thatS" erf (x) dx =o CA

A: ddx(erf(x)) =2 e-x2π   I =∫-aa erf(x) dxI= erf(x) x -∫ ddxerf(x) x dxI = x erf(x) -2π∫ x e-x2 dx-x2…

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

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: Functional theory and functional analysis

A: According to the given information it is required to prove that:

Q: Let x, y ∈ R such that x 0.001. Prove that |x · y| = |x| · |y|

A: Given: x, y∈ℝ, x<-0.001 and y>0.001   To prove: x·y=x·y

Q: 8.F. Ifx= (1, I, --- ,)€R", define x, by e that z- l, is a norm on R'. Prove

A: Given that x=x1,x2,..,xn and x1=x1+x2+...+xn. The objective is to find the whether x1=x1+x2+...+xn…

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

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Q: 2/Define the topology If X-R² and T = {GK, KERZULAR} GK = {(XY) ER²|x+y >K3 Prove is topolgy on R²

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Q: Let po) - X In (x) , using the following approx im for the secoud denvative fcxo) = FOXO -h)-ZR…

A: Introduction: Apart from differentiating a function analytically, the derivative of a function can…

Q: 1) Prove that Ym,n EN such thert m¢ nt = n<m using the princdple of in duction!

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Q: prove that erf (x) dx =o

A: To prove: ∫−aaerfxdx=0

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

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Q: Prove that Z(Sn) = {e} for n ≥ 3.

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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: Proof by contrapositive that for x, y ∈ Z, if xy is even then either x or y is even.

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Q: Prove Theorem (AT )T = A

A: Given: Theorem, ATT=A.

Q: Prove that Theorem 13.1 (vi) using the simple ftn, nonnegative ftn and general ftn. (vi) If X is…

A: vi) From the given information, X is measurable in N.  There exists a sequence of simple functions,…

Q: How to solve this?

A: We are given that x + y + z = 5 where x, y, z ∈ R. Now, we need to prove that x2 + y2 + z2 ≥ 5/3.…

Q: 4. Let xn → 0 and (yn) be bounded. Prove xnYn → 0.

A: Hello, thanks for your question but according to our guidelines I am answering just question number…

Q: Let f(z, y, z) = z2 +y? + z²,z = /utv, y = /u – v, and z= uv, Find in terms of u and v.

A: we have to partial derivative with respect to u by using chain rule

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: 2. What is the largest possible c in (1) if X=R² and x₁=(1,0), x₂ = (0, 1)? If X = R³ and x₁ = (1,…

A: Given: c To find: a) X=R2b) X=R3

Q: (3) In R² with the standard topology: {(x, y) | x² + y² {(x, y) | x² + y? > 1}. = 1}, {(x, y) | x² +…

A: In a topological space R2with standard topology. A  subset O is open if at each point x in O there…

Q: Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x,g (y) = 3y + 4 and h (z) =…

A: Given, Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x, g (y) = 3y + 4 and…

Q: If z = f(u) + g), where u = = and v = x· y. Prove that y x: Zx - y Zy = y? Zyy – x2 - Zxx using the…

A: We have to prove this

Q: For x, y, z, w > 0, prove that (x/z)+(y/z)+(z/w)+(w/x)≥4

A: It should by x/y, instead of x/z

Q: Prove by direct method: if x, y>0  such that x ≤ y then √x≤√y

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Q: If x = v'u and y = 2uv, show that JJ' - 1, where J is the jacobian of u,v with respect to x and y…

A: We have find the jacobian

Q: prove that Sa/Gn has f discription with M-ly ny-l Degrees tof freedom

A: Suppose that xi and yi are two random samples of size n, drawn from a normal population.  Population…

Q: In general, for a a) S is compact 2) Sis closed at S is compact

A: We use definition of compact metric space, totally bounded set, Closed set, accumulation point or…

Q: Show that, with z = Rw, for some bounded w, || z(jw)|| w llw(jw)| ||R||. = sup means that -

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Q: Suppose f" exists and is bounded on Co,0), Prove that if bam fa)=0, then bim fGx)=0.

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Q: Prove that (i) Res (tan z, r/2) = - 1

A: Here,we have to prove that Res(tanz,pi/2)=-1

Q: Show for all x, y in R |x+ys]xl + ]y!

A: consider left hand side  and use the result x=x2 x+y=x+y2=x2+y2+2xy≤x2+y2+2xy≤x+y2=x+y

Q: Theorem:For all Xiyiz,w EN i) X<y ^ z sw =)x+Z <ytw ii) xsy ^ zw =) x+Z< y tw )x<う tsw 1 Esw うメ+ と x…

A: Solution: We know that for x,y∈ℕ If x&lt;y then ∃k∈ℤ+,k&gt;0 such that x+k=y If x≤y then ∃k∈ℤ+,…

Q: If G (x, y, z) = curlF (x, y, z), where F (x, y, z) = (y – 2, z – x, x – y), then х, х — у), then ||…

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Q: Let fun chan defined by g(2) = V Z+8' –-8, ts devivalie z+h+8- Vz+ 8 is equal to the expressian. A)…

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Q: dv 4.)Prove that f -Incosv + Insinv + C sinvcosv

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Q: Let x, y, z ∈  ℕ. Suppose gcd(x, y) = 1. Prove that if x | yz, then x | z

A: In the given question we have to prove that if x|yz , then x|z and g.c.d(x,y)=1 and x,y,z∈ℕ. we will…

Q: A mapping f:C→ C is called bounded, if sup |f(2)| < o0. zEC Show that sin, cos : C →C are unbounded…

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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: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

A: