Prove the given statements by the indicated method. (1). By Direct Proof: "Vm, n € Z+ - {1}, m² | n² → m n."

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A: Detailed explanation mentioned below

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Q: Let 0 E A CR. Show that if A is both open and closed, then A = R. (Hint: Let M = sup{t: [0,t] C A})…

A: We prove this by contradiction.

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A: It is given that fx=ax+b and we need to calculate fx+h-fxh.

Q: V3n V3n + 4 + 3n² V3n =? n2 + 3n² lim ... n→o 1+ 3n2 A) pi/6 В) о O C) NONE D) 1 O E) pi/3

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Q: Prove that : L²₁ (1-x²) Pm Phd x T 0. 2n (n + 1) 2n + 1 4 mn. m = 11. m, ne N

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Q: If T: R"„8* is a Linear transfor matiom and m (x)= (2² +2)Cx+ 3. Find all the ratinal canomical ferm…

A: Companion matrix:  Let f(x)=xn+an-1xn-1+………+a0 be a polynomial then the matrix form is given by…

Q: 1 О. 15 Show that %3D (D - Q) k e k

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Q: {[(B U C)U A] N [(B UC)U A]}n {[(A U B)UC] nÃ}

A: We use the following properties of set operations: Distributive law: (A∩B)∪C=(A∪C)∩(B∪C)…

Q: PLS HELP ASAP ON ALL ASKED QUESTIONS PLS

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: 3.3. Let a, BEC. Prove that la + B2 + la – B|2 2(la|? + |B|?), and deduce that |a + Va? – B2|+ |la –…

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Q: Let A = 0 00 G|5 0 0 0 , u = 1|5 15 10 - 25 and v= g d h Define T: R³ R³ by T(x) = Ax. Find T(u) and…

A: Let and ,Define Then,   and

Q: (c) Let z1 = 2+ 2i and let 22 =1+i. Prove that the triangle inequality |21 + z2| < |21| + |22|…

A: Left hand side is equal to, z1+z2=2+2i+1+i=3+3i=32+32=32 Right hand side is equal to,…

Q: 2 Find.all.se.cond.partial.devevatives:. f(x,y)In(3xi+5.y..b.ue sint C.Z-ytan x a..

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Q: Q73. If P(A n B) = and P(A' n B') = ½³, P(A) = p and P(B) = 2p then value of p is :

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Q: . True or false? In each case, provide a proof or a coun- terexample. (a) ACC, B CD→ A x B CC x D.…

A: Set and subset true false statements

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A: Introduction: When identical statement variables are used to denote the component statements, two…

Q: Please

A: We know that, Statement: If P then Q. Contrapositive: If ~Q then ~P.

Q: u = 3, o2 = 9, P(-6 < X<12) a. Determine P(u – ko < X<u+ ko) 1 - 1/k?

A: According to the Chebyshev's theorem, 1-(1/k²) of the observation fall within k standard deviation…

Q: Which of the following is true? * If A {x: x < 5,xE N} and = B = {y: y? + y - 2 = 0,y € R}

A: Solution:

Q: Let q E Q and x E R - Q. Prove that q + XER - Q.

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Q: 2/3 2/3 Sindangquationaflatangat linotothe cunse X (8,-4) " ye 2 a0at the point

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Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

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A: We know thatcos(x) = ∑k=0∞-1kx2k2k!

Q: O Find the solution of the linear System by Uring A', X, +2 X2+3X3 - 5 2X, +5 X2 + 3 X3 3 übouri X,…

A: Given equations:  x1+2x2+3x3=52x1+5x2+3x3=3x1+8x3=17 Writing this system of linear equation in…

Q: 1 Prove that V n 2 1. j=1 j(j+1) (n +1)

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Q: 9. Prove or disprove. (a) Vn e Z*, n² < 2". (b) Vm e Z, if m is odd, then m² is odd.

A: In this question we have prove if this equation are valid or not.

Q: Problem 2)-t): Let Cn,d be the number of integers with 1 ≤ m≤n such that gcd(n,m) = d. Explain why…

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Q: Which of the following statements is true? 1. (A) = (AT)" for every positive integer n. II. If A is…

A: To determine whether the given statements are true or not.

Q: Let 0 E A C R. Show that if A is both open and closed, then A =R.

A: Given that Let 0∈A⊂R.Show that if A is both open and closed, then A=R. Here we use the principle of…