Let a, y, z be positive real numbers such that r +y+z xyz. Prove that 1 1 1 1 1 1 1+ x2 1+ y2 1+ 22 1+ xy 1+ yz 1+ zx

Q: Find real numbers x and y such that iy 2 - i i + 3 1 + i ||

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Q: Assume that a b•c and that gcf(a,b) = 1. Prove that a lc (hint: use the result that gcf(a,b) = 1 iff…

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Q: Prove that if R and S are integral domains, then the direct sum R ⊕ S is not an integral domain.

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Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

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Q: 2. Prove the general formulas (GF)» = G«F* and (F-')» = (F«)- in the special case where F and G are…

A: Given G and F re isometries in ℝ3 To prove  1.GF*=G*F*2. F-1*=F*-1 Since F and G are isometries in…

Q: 23. Prove that if D is an integral domain, then D[x] is also an integral domain.

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Q: Prove: ∀R x>0 and ∀Z n > 0, there is exactly one positive real y such that yn=x.

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Q: (1) Prove that the product xy of two real numbers x and y is nonnegative if and only if the absolute…

A: First let xy be positive  Then either both x and y are positive or both are negative  If both are…

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

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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: Let R be an integral domain with unity and let r, s ∈ R. Prove that if ⟨r⟩ = ⟨s⟩, then r and s are…

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Q: 4. Prove that (À x B) - (B x C) x (C x A) = [AB %3D

A: The solution is given below.

Q: 2. Prove the general formulas (GF)* = G*F* and (F')* = (F*)' in the special case where F and G are…

A: An isometry in R3 is a one-one onto function which preserves distance that is for a ∈R3and b ∈R3…

Q: 1- iz If z = x +i y and w = z-i show that |w| = 1-z is purely real

A: i2 = -1

Q: Prove that the quartic function q(z) = r* - 16 has only

A: Given that: q(x)=x4-16

Q: a+ib a) If x+ iy = prove that x? + y² = 1. a-ib

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Q: 2) Let R' real numbers, define = x,Y1 + 3x2y; + 5x3y3 for x = (x, X2, X3) and y (y,Yz, Y3). Is…

A: Given x=x1,x2,x3, y=y1,y2,y3 in ℝ3. Define x,y=x1y1+3x2y2+5x3y3

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: 2) Let R real numbers, define = x,Y1 +4x2y2 + 6x3Y3 for x = (x1,x2, X3) and y (y1,y2, Ya). Is inner…

A: Let V be a vector space over ℝ.An inner product <·,·> is a function V×V→ℝ with the following…

Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express z x and zy in terms of fu , f y, and the…

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Q: Prove that f-(U B: -U(B.), %3D i=1 i=1 and n N (B.). 3=D1 i=1

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Q: f:Z → Z, f (m) - Зт - 2 g:R → S, where S = {x E R|x 2 0} g(x) = |x| h(x) = In(x) R*is a set of…

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Q: Prove: if z = (V3)/2 + i/2, and %3D then z = i z" = 1 %3D

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Q: 5. Let f(z) = 3z* + iz3 + z + 7. Prove that all roots of f (z) are in the annulus 1 < |z| < 2.

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Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

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Q: Let A- ] and ů = (3) %3D %3D Commpute Av Solve for x 2-x det 10 1- X

A: Use matrix multiplication, to calculate Av⇀. Av⇀=2310113=21+33101+13=2+910+3=1113 Therefore,…

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

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Q: Find all valus of z such that (5 + z)6 = (1+ i)z6

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Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

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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: Prove that there are infinitely many triples (x, y, z) of positive integers satisfying x² + y? = z*…

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Q: #5 Let x, y, z be nonnegative real numbers such that y + z> 2. Prove that (x + y + z) >4x + 4yz.

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Q: Let ū = 2i – k and ī = i+3j + k. Compute |ū], |T], ū x ī, and Projzū.

A: Solution is here

Q: Let f : A → B and g : B → C be bijections. (a) Prove that (go f)-1 exists. (b) Prove that (g o f)-1…

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Q: 2-Express each function in the form u(x, y) + iv(x, y), where u and v are real, (c) ez2 (d) In(1+…

A: We have to write in u(x,y) and v(x,y) form.

Q: p(z) Resz=z0 q(z) p(zo) q'(20)

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

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Q: Show that the functions x and x2 are orthogonal in P5 with inner product defined by (5), where xi =…

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Q: Let R→R be defined by g(x) 2y+1 2y+2 Is g bijective? Prove or give a counter example.

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Q: 22. (a) Prove that [0, x] = [0, y] for all x, y E Z, x + 0, y # 0. (b) Prove that [ax, ay] = [x, y]…

A: (a) we have to prove that  0, x = 0, y for all x, y∈Z, x≠ 0, y≠0 proof as x, y∈Z, x≠ 0, y≠0 Z is the…

Q: Let an 1+1/V2+1/V3+...+1//n , Prove {an} is unbounded. %3D

A: Given that, an=1+12+13+...+1n

Q: Let x, y and z be real numbers. If x > y and y = z, then x > z. For any real numbers x and y, with x…

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Q: Use the generating function to prove that J,(x + y) = J,(x) J-,). %3D n=- 00

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Q: (a) Evaluate the integral [21,22,23,21] 3(z − 1)²dz, z₁ = 1, %2 = 1 + i, z3 = −1 + i. (b) Show that…

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Q: Prove that B (xX+1; y ) + B (x,y+1)=B (X,Y) let П (х+1) = ХГ (Х)

A: We have the formula for beta function interms of gamma function is that B(m,n)=Γ(m)Γ(n)Γ(m+n)As we…

Q: a) Check that the functions e^x, e^-x,coshx are solutions of y'''-y'=0 b) show directly that…

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Q: Let ƒ : [0, 1] → R be given by f(x) = x and g : R → [0, 1] be given by g(x) = Prove that |[0, 1]| =…

A: Cantor-Schrӧder- Bernstein Theorem: if there exist injective functions f: A → B and g: B → A between…

Q: 5) Let z = f(x2 - y²). Show that y = 0 ду %3D ax

A: Here, ∂ represents the partial derivative, which means when differentiating with respect to x,…

Q: (18) (a) Prove that R* is isomorphic to R>0 x Z2. (b) Is Q* is isomorphic to Qo x Z2? Prove your…

A: a)  To prove :- ℝ*is isomorphic to ℝ&gt;0×ℤ2. Proof :- Define a map ϕ : ℝ&gt;0 ×ℤ2→ℝ* by…

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.