Theorem 4.1. if r, s ER and r

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: . Prove that an→L = Jan] → |L]. (Make cas

A: Given that an is a sequence which converges to L that is, an→L. Then we have to prove that |an|…

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: 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: Prove that if A and B En and Ac B, then

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Q: 2. (6 marks) Let a, b, c € Z with a 0. Prove that if a X (a + 2b + 3c), then a Xb or a Xc.

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Q: Prove that R(5, 5) < 50.

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Q: 2h 1+h )-2 = 1 – if h<<r Prove that ( %3D r

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Q: Prove that if r>3 and y > 4, then ry > 11.

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Q: Evaluate |/I. ze*+ydV where B is the box determined by B 0 <x < 4, 0 < Y < 3, and 0 < z < 3. The…

A: We have to evaluate the integral

Q: et x + y+ z=5 where x, y, z E R. Prove that x² + y² + z² >

A:

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: If a € R such that 0 0, then show that a = 0.

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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

Q: 1. define For R", one can consider the following norm structures. For all x = (1,,n) R", we • ==…

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Q: 3.) dy_x²(1-y) = 0 dx

A: To solve the following 1st Order Linear DE.dydx-x21-y=0

Q: 9. Which of the following are true for all sets A, B, and C. If the statement is true, prove it by…

A:

Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: 2.) dy dx 1 x²+1

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Q: Fin Sup(S) and Inf(S), Where 1 S = {1. п

A: Consider the set S=1, 12, 13, ... , 1n, .... Observe that as the value of n approaches to infinity…

Q: 6. Let A = {x|x € R and x² - 4x + 3 < 0} and B = {x|x € R and 0 < x < 6}. Prove that AC B.

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Q: "for all n,m ∈Z \ {0},  gcd (n,m)= 1 then there are a,b ∈ Z with an+bm= 1" is the converse of the…

A: It is given that if,  gcdn,m=1 then there exists a,b∈Z such that an+bm=1 To show that converse of…

Q: 4. Show that R¹ and R² are not homeomorphic. How about R¹ and Rn for n ≥ 2?

A:

Q: S6 Let X, .. Xn be a r.s. from N(0,0). Show that T, = X, is not a complete sufficient for 0 but T, =…

A: Given Information: X1,X2,.....,Xn is a random sample  from a Normal distribution, N(0,θ) . It is…

Q: 1.) dy dx + y = cos(x) = 0

A: Let's solve given differential equation.

Q: 7.) Evaluate F dř. C F(x, y, z) = (3x, 4y, xz) r(t) = (6t+2, 8t - 3,7t), 0≤ t ≤ 1

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Q: If p,q,r are prime number but p is not equal to q, then what is gcd(p,q)? What is the smallest…

A: It is given that p and q are prime numbers. Assume that gcdp, q=d. Then, d is the greatest common…

Q: |U( Z O Z O Z)| is: O 2 16 4 K O O

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: "for all n,m ∈Z \ {0} and g ∈ N,  gcd (n,m)= g then there are a,b ∈ Z with an+bm= g" is the…

A: We need to see if the converse of the statement "gcdn,m=g then there are a,b∈Z with an+bm=g" is true…

Q: 4. Find the number of surjections from {1, 2, ..., m} to {1, 2, .. ., n} such that no element in the…

A: We have to find the number of Surjection from one set to another set.