S = {x2 − y2 ∈ N : x, y ∈ Z, x2 − y2 > 127}. Prove that S has a smallest element.

Q: 9) Use Fubini's Theorem to compute the double integral ƒƒ_ƒ(x,y)dª_where…

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Q: The set Qn (0,2) has Lebesgue measure 0.

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Q: (b) Let x € R. Prove that if x² – 2x − 3 ≥ 0, then x² + 1 > 0. -

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Q: b) Let E = {0,±3,+6,...} be the set of integers that are divisible by 3 and let T = {0, +5, +10,...}…

A: Explanation of the answer is as follows

Q: Use the change of variables X1 X2 X3 X4 = = G(r, 01, 02, 03): = to compute the content of the 4-ball…

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Q: The value(s) of x that satisfy In(x² + 1) – 2 In(x) = In(16) – In(8) are/is

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Q: Prove that x(t) = xoe At xoe At is unique solution

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Q: Let S=X belongs to R: x> =1. a) Show that the set S has lower bound, but no upper bounds. b) Show…

A: The answer is given as follows:

Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: If X is uniform in the interval [-3,3] compute X(i.e. E{X}) and o² (i.e.E{(X-X)²}).

A: Given that X ~ U[-3,3] to find mean and variance

Q: assume that Pr[A]=0.25Pr[A]=0.25, Pr[B]=0.35Pr[B]=0.35, and Pr[A∩B]=0.1Pr[A∩B]=0.1. 1)Pr[A|B] =…

A: Given: P(A) =0.25 P(B)=0.35 P(A∩B) =0.1

Q: Find real numbers x, y and z such that (x + 4)i + (y – 5)j + (z – 1)k = 0

A: To Determine :-  The real numbers x , y and z  such that   x + 4  i +  y -5  j +  z - 1  k =0

Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r>d/2 where x, y, z belongs to Rn So locus z…

Q: Prove or disprove: If (n) and (yn) are subsequences of each other, then n = yn.

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Q: Find a ≤ b such that

A: ∫abx2-9dxApply the Sum Rule:…

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: Let Z - N(0, 1). Find P(Z < 2). Question Help: D Post to forum

A: Given that, Z~N(0,1)

Q: For x,y E [0,1], prove that xy(1-x-y)≤1/27

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

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Q: (2) Prove that for all x, y € R, we have 8xy³ ≤ (2x + y³)².

A: A.M&gt;=G.M

Q: Let G = &lt;z | z6 = 1l and H = &lt;x, y | x2 = y3 =1, xy =yx&gt;. Showthat G and H are isomorphic.

A: Given :  G = z : z6 = 1 and  H = x, y : x2 = y3 = 1, xy = yx To show : G and H are isomorphic.

Q: Show that the characteristic of Z[i]/&lt;a + bi&gt; divides a2 +b2.

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Q: 1. Translate these statements into English, where the do- main for each variable consists of all…

A: a) for all x, there exists y such that x is less than y b) for all x, for all y if (x is greater…

Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: S 1.Prove that L (coswt): %| s2 +w2

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Q: Show that |Re(z)| + |Im(z)| ≤ √ 2|z| for any z ∈ C

A: We will use the basic knowledge of the complex number system to answer this question.

Q: Find gcd(2,3+5i) in Z[i]

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Q: Suppose AE R, v#0, and (A- A)v#0. Use the fact that A is symmetric show that it is impossible to…

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Q: 1. Determine whether each of these functions is a bijection from R to R a) f (x) = 5x + 10 B) f (x)…

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Q: Prove, without using any results from the chapter notes, that for all z = C, z² = 2².

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Q: 4. Prove that, VR, -11-

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Q: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

A: Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

Q: For each subfield S of Q[i, z], determine Auts(Q[i, z]).

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Q: Ifx.y,z are real numbers satisfying the equation 25(9x + y) +9:*–15(5xy + yz + 3:x)= 0 then prove…

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Q: Prove that for any x > 0, y>0 (x² + y²) ³ ≥ (x³ + y²³) ³. Hint: Introduce a new variable z = (2)².

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Q: Suppose u, v belong to V. Prove that = 0 if and only if ||u|| ≤ ||u+av|| for all a belong to F

A: We have to prove  if and only if