Let A = {: : neN, mN, m >r m (a) Prove that inf A 0. (b) Prove that sup A = 1.

Q: Q2 (a) Let n E N and let 7, w e R". Show that if ở - = 0, then || † + w||? = || T||? + ||||?. (b)…

A: To answer the given question we shall use the following rule: 1.   If x→∈ℝn, then…

Q: Prove by bijective (one to one onto) of function 2" f: 2ln] P(({0, 1}, n)).

A: In this question we have to deal with the cardinality of a function.

Q: Prove that Vn > 0: 2º + 21 + .…+ 2" = 2n+1 – 1. ... an+1_1 Prove that Vn > 0: aº + a² + .….+ a" =…

A: Here, the objective is to prove the following: 1) 20+21+22+......+2n=2n+1-1   ; n≥02)…

Q: 2.a) Find Vop if p = In|r]. b) Find Vp if p==. c) Show that V7" = 1.-27. d)Prove that V² (¹) = 0.

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Q: -w #43 Let n», ², a; >0, 5=1,2,... and a₁ + a₂+...+ an = 1 Prove that a, a ₂ 2-92- + 4 + an 2-an 7…

A: We need to prove summation.

Q: on Give an crample of a dis continous bijection fo[at] = [1,2]

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Q: Find the value of the Conj uction g(+) ifya know that w(f.g)=3e" and fCo=et (Wronskain)

A: We will find out the required value of g(t).

Q: prove the ged (atb,a?+b³) =16Y2 assuming that gcd (a,b) =1 •

A: It is given that gcd(a,b)=1

Q: Find the fowien senies fon | X), -T<X <A

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Q: A) Show that the surtace x-2yz ty' =4 is orthogon al to any of the sm tamily of sur faces…

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Q: What is u × w, assuming that w × u = 〈2, 2, 1〉?

A: Given,w×u=2,2,1 To find:u×w Solution: We know ,by property of cross product of two vectors (say,a…

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: If Z- N(0,1), then P(0.33<Z<1.05) is: a) b) 0.129 0.353 c) d) 0.224 0.482 Accesibilidad todo corredo

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

Q: Prove that i 224 s 2xP for xzO, y zo i and 1 t L-l with p>

A: Given that x,y≥0. And 1p+1q=1 with p&gt;1. Consider gx=xpp+1q-x. Then g'x=xp-1-1 and g'x&lt;0 when…

Q: 4. a) Prove that i) V=1 ¡¡) fx+nh = Σk=0 I A I+A 1+A (1) Ak fx.

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Q: Qt Let T= {carb) a,b ER\fopp and of defined onI by Carb)*(c,d)= (ac, bd) for all (a,b), (c,d ET,…

A: The set T is defined as, T=a, b | a, b∈ℝ\ 0 The operation * is defined on T as follows, a, b*c,…

Q: lemma: let f X→2, cont. J I' yCY with Subspace topalagy and then Fly :Y-→ (restrection of Ĥ to Y)…

A: Result: is continuous iff and only if f-1(V) is open in X whenever V is open in Z. Given:  is…

Q: (b) If WCU, prove that (Recall that U+W = (c) If U + W = U, prove that WCU. U + W = U. = {u+w|u€ U₁w…

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Q: Vove erf C-∞)= -I or J edt=-1 STT

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Q: which of the following statements is for t*t, the convolution oft with t" Select one: а. S. T(t –…

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Q: Q4. Let A E Rnxm, n ≥ m. a) Use the SVD of A to deduce the SVD of ATA. b) If m = n and A is…

A: Given: The given expression is, A∈ℝn×m , n≥m a) To use the SVD of A to deduce the SVD of ATA b) To…

Q: Let x, y ∈ R. Suppose x2 + y2 = 0. Prove that x = 0 and y = 0.

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Q: 6x:-ld 212CK) 's aving of integer num 12 and fo), gaJE Z, ) such That 9Cx) = Prove Fhat dey (fagx))…

A: Given  Ring Z12(x)  f(x) = 2+5x+4x4  g(x) = 1+x+x2+3x3    So clearly,  degree of f(x) = 4  degree of…

Q: 2 1] Assume A = 1 3 4. [2 2 4 and B = show that (AB)' = B' A' [3 6 4]

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Q: Q Let 8=R and Prove disprove that zis atopdagy or on R.

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Q: Prove that dv du 2 – u? + v² 18

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Q: (1) Let F =x?++zÂ, and let u= ||FlI. Let fcu) = 3ue 2-2u Show thet Df(u)= 6-6u)eu F -2u

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Q: If S:= {1/n - 1/m²n,mEN }, find inf Sand sup S.

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Q: let f [o2]→Rbe afunction if x<1 fex)= Is f Riemaninteglable? 下くx か

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Q: 1(42). M = "+ x/F + 78 +1=24 S

A: According to answering guidlines we are doing first question if one or more question given

Q: prove that in(a+b) ~ (nat (a), when bica

A: ln(a+b)=lna1+ba, where b≤a=&gt;ba≤1 ln(a+b)=ln(a)+ln1+ba by using the property of…

Q: Let A = { n - m : neN, me N, m > n}. (a) Prove that inf A = 0. (b) Prove that sup A = 1.

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Q: let 19, b) = 1, them prove that there exists x20 Such that ijzo ax_by = 1

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Q: Sinx n to 5. Airod ht Let fix)=) Find fio) ) and f"to)

A: According to question given thatf(x)={sinxx,x≠01    ,x=0

Q: ) Prove that Вiт.m) - I(z)(m) for m >0 22'T(m +) 2

A: We use the property of Beta function and relation between Beta and Gamma function to prove the…

Q: Consider the following statements: P : x is odd, Q:x is positive, R:-4< x < 3, S : |x| < 2. Find all…

A: See the detailed solution below.

Q: Consider the following statements: P:x is odd, Q: x is positive, R: -4 <x < 3, S: æ| < 2. Find all…

A: The given statement is a logical if-then statement. So this statement becomes false only when S is…

Q: Let -J-6:30 = (3 1 A=-2 b= Then Lb is the unique solution of Ax=b.

A: We are given .Here, we need to verify that Lb is the unique solution of Ax=b. So, if we can verify…