If MC Z is a subset closed under addition and subtraction, then =dZ for some d € Z satisfying d≥ 0. `a, b € Z, then aZ + bZ = dZ and aZ ^ bZ = lZ, for suitable d, l = Z oes (aZ) (bZ) = abZ? Why is az U bZ of less interest here?

Q: x2-a? A B X-a x+a Find A, where a + 0. O a -- a

A: Using partial fraction,

Q: 9. If (X, M, ) is a measure space and E, FEM, then μ(E)+μ(F) = μ(EUF)+ μ(EnF).

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Q: When I was considering what to do with my $10,000 lottery winnings, my broker suggested that I…

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Q: 2. Find all volves of tand u x(t,4)=0 and simultomebusly such that =0. 3. 5x-11* C.

A: Get the equations for:  fx=0fy=0 15x2-11y=0-11x+3y2=0 or 15x2=11y3y2=11x

Q: whats the mumber of injections f from S={1,2,3,4,5} to T={a,b,c,d,e,f} such that f(1)!=a and f(2)=!b…

A: We are given that we have two specific elements in the domain S which must map to specific elements…

Q: Suppose V is finite-dimensional and v € V with v ‡ 0. Prove that there exists € V' such that p(v) =…

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Q: Suppose you are given four natural numbers w, x, y,z e N such that w>x and y > z. Show that ly – w|…

A: It is given that w, x, y, z are natural numbers such that w>x and y>z. Now we want to show…

Q: 4. Evaluate [In(z+i)dz along the line from 1 to 1+i3

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Q: b) Describe the points at which the function describes the different pieces of the function. c) Is…

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Q: 5. Let (X, B(R), µ) be a Borel measure space. Suppose that {An = ["+1, 4n-1]}, be a family of…

A: Let (X, B(R), µ) be a Borel measure space

Q: 19. The real number k lies between 0 and 1. The area of the quadrilateral formed by the lines y =…

A: The objective of the question is to find the area of the quadrilateral formed by the lines y = kx, y…

Q: 1. re (x, y, z) x). Ly). (z}} - *(*. y, z) . (x)S(x), ly), (z]} (x)E (x, y, z}

A: If x is element then it belong to set And {x} it will be subset of set

Q: Expand: y= In O y = 6 lnx – 4 In w+3 Iny- In z Oy =6 Ing – 4 In w – 3 Iny + In z Oy =6 Inz +4 In w -…

A: We will simplify the logarithm.

Q: الرابع: جد ناتج احد التكاملين التالي ين 1- Smte) xy e*y° dxdy 2- r? N4-y² ry8-x²-y" zdzdxdy x²+y2 2-…

A: 1 ∫01∫0ln(2)xyexy2dxdy first we will solve ∫0ln(2)xyexy2dx ∫0ln(2)xyexy2dx=y∫0ln(2)xexy2dxSubstitute…

Q: • If f(x) = |sgn(x- 1)| then, 1. D; = R RF [1,0, –1} 2. D; = [1,00) R; = {1,0} 3. D; = R Rp = {1,0}…

A: Given that, f(x)=sgn(x-1) Take the derivative for f(x), It is known that, D(sgn(x))=δ(x-1) Then Df…

Q: Please help me with this question. I am having trouble understanding what to do. Please show all…

A: The objective of question 3 is to find the largest set of values n for which 2^n < n! and prove…

Q: f (u, v, w) can be differentiated and if u = x-y, v = y-z, w = (z-x) ôf of ôf + + ду дг

A: Since is a function of and they are described as, Thus, we can say that is a composite…

Q: 9. If (X, M, u) is a measure space and E, FEM, then u(E)+μ(F) = μ(EUF)+ μ(EnF).

A: Introduction: A measurable function is a function between the underlying sets of two measurable…

Q: 5.2 Any lower Lebesgue-Darboux sum does not exceed any upper Lebesgue-Darboux sum i.e. for any two…

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Q: Evaluate fry²dx + x^dy, where C' is the rectangle with vertices (0,0), (4,0), (4,1), and (0,1).

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Q: 14) Find the Jacobian of the tra nsformation. x=5u-v,y=Du+3v

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Q: 6. Let U = {2,4,5,6,8,9,10} Y = {yly = 0.5U} Enumerate: a.) P(X U (YOZ)) X = {x | x² 13x + 42 = 0} Z…

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Q: Let 9,n be antegers wth 2n and ged (a,n)=1. Show tht of U(Z.) ={a*+nZ:tez} then o#1 mod 6) for all…

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Q: If fufe.- fuave diffa Veubable at'xo eI fuave difle Veubable at Xo eI Pi:I R,ists-n prove O…

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Q: Q/ The center of agrouP G is denoted by xCG) one of the following Statements is false: O Z(Sn)={e}…

A: Let G be group and the center of G is denoted by Z(G) and is given by ZG=x∈G:ax=xa,∀a∈G that is…

Q: Problem 12 Selve haplace is equation +31 (124) + 1/2 3/4 u for 1 (1₁0) on the half-disc 04542,…

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Q: If X₁ and X₂ are independent rv's then Px₁+x₂ (t) = x₁(t)¶x₂(t) More, generally for independent rv's…

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Q: 1. v₁ = (4, −2, 6, −4), V₂ = (6, −3, 9, −6)

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Q: (b) If A is naturally increasing, then A is not spottily decreasing. This statement is O True O…

A: note :As per the instructions we will only solve part (b) of the question.As per the question, we…

Q: If p(E) =²/₁ find o(E) and o(E'). 9 o(E) = o(E¹) =

A: O(E) stands for odd of events

Q: 15. Suppose that a, b e Z and gcd(a, b) = 1. Prove that ged(a", b") = 1 for m, ne N. N and lot 8 frm…

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Q: • If f(x) = then, 1. D; = R 2. D; = [1,00) Rf = [0, 00) 3. D; = R R, = {1,–1} 4. D; = [1,00) R =…

A: Note that fx is not defined for x=0

Q: R and r are defined by E(r, – 5)(y – T) Sry VS., Syy SSR R2 r = SST' (1) Show R? = r².

A: If y = a+bx is the linear equation  Correlation attempts to measure the strength of the relationship…

Q: 2. Let A € M, (R) be given. (a) If AA = 0, prove that A = 0₁. Does this hold if A € M₂, (C)? (b)…

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Q: 3. Prove F (Vr)(A→ B) → (Vx)A → (Vx)B. ->

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Q: 12. Prove: ΔUΧΖ-Δ AUXZ AYVZ ~ Z.

A: Given: chord UXChord YVand the Center ZTo…