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Question Number 191786 Answers: 0 Comments: 0

Ques. 1 (Metric Space Question) Let X = ρ_∞ be the set of all bounded sequences of complex numbers. That is every element of ρ_∞ is a complex sequence x^− ={x^− }_(k=1) ^∞ such ∣x_i ∣<Kx^− , i=1,2,3,... where Kx is a real number which may define on x for an arbitrary x^− ={x_i }_(i=1) ^∞ and y^− ={y_i }_(i=1) ^∞ in ρ_∞ we define as d_∞ (x,y)=Sup∣x_i −y_i ∣, Verify that d_∞ is a metric on ρ_(∞.)

$$\mathrm{Ques}.\:\mathrm{1}\:\left(\mathrm{Metric}\:\mathrm{Space}\:\mathrm{Question}\right) \\ $$$$\:\:\:\:\:\:\:\:\mathrm{Let}\:\mathrm{X}\:=\:\rho_{\infty} \:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\: \\ $$$$\mathrm{bounded}\:\mathrm{sequences}\:\mathrm{of}\:\mathrm{complex}\: \\ $$$$\mathrm{numbers}.\:\mathrm{That}\:\mathrm{is}\:\mathrm{every}\:\mathrm{element}\:\mathrm{of} \\ $$$$\rho_{\infty} \:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{sequence}\:\overset{−} {\mathrm{x}}=\left\{\overset{−} {\mathrm{x}}\right\}_{\mathrm{k}=\mathrm{1}} ^{\infty} \: \\ $$$$\mathrm{such}\:\mid\mathrm{x}_{\mathrm{i}} \mid<\mathrm{K}\overset{−} {\mathrm{x}},\:\mathrm{i}=\mathrm{1},\mathrm{2},\mathrm{3},...\:\mathrm{where}\:\mathrm{Kx} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{which}\:\mathrm{may}\:\mathrm{define} \\ $$$$\mathrm{on}\:\mathrm{x}\:\mathrm{for}\:\mathrm{an}\:\mathrm{arbitrary}\:\overset{−} {\mathrm{x}}=\left\{\mathrm{x}_{\mathrm{i}} \right\}_{\mathrm{i}=\mathrm{1}} ^{\infty} \:\mathrm{and} \\ $$$$\overset{−} {\mathrm{y}}=\left\{\mathrm{y}_{\mathrm{i}} \right\}_{\mathrm{i}=\mathrm{1}} ^{\infty} \:\mathrm{in}\:\rho_{\infty} \mathrm{we}\:\mathrm{define}\:\mathrm{as} \\ $$$$\mathrm{d}_{\infty} \left(\mathrm{x},\mathrm{y}\right)=\mathrm{Sup}\mid\mathrm{x}_{\mathrm{i}} −\mathrm{y}_{\mathrm{i}} \mid,\:\mathrm{Verify}\:\mathrm{that} \\ $$$$\mathrm{d}_{\infty} \:\mathrm{is}\:\mathrm{a}\:\mathrm{metric}\:\mathrm{on}\:\rho_{\infty.} \\ $$

Question Number 191775 Answers: 2 Comments: 0

Question Number 191758 Answers: 2 Comments: 0

Question Number 191756 Answers: 2 Comments: 0

Question Number 191744 Answers: 0 Comments: 0

Question Number 191735 Answers: 3 Comments: 0

Verify that ┐(p→q)→(p∧^┐ q) is tautology using laws of algebra

$${Verify}\:{that} \\ $$$$\:\urcorner\left({p}\rightarrow{q}\right)\rightarrow\left({p}\wedge^{\urcorner} {q}\right)\:{is}\:{tautology}\:{using}\:{laws}\:{of} \\ $$$${algebra} \\ $$

Question Number 191733 Answers: 2 Comments: 0

Show that lim_((x,y)→(0,0)) ((x^2 −y^2 )/(x^2 +y^2 )) does not exist

$${Show}\:{that}\: \\ $$$$\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\:\:\:\frac{{x}^{\mathrm{2}} −{y}^{\mathrm{2}} }{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:{does}\:{not}\:{exist} \\ $$

Question Number 191732 Answers: 0 Comments: 0

Find the relative maximum and minimum of the function f(x,y)=x^3 +y^3 −3x−12y+20

$${Find}\:{the}\:{relative}\:{maximum}\:{and}\:{minimum} \\ $$$${of}\:{the}\:{function} \\ $$$${f}\left({x},{y}\right)=\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} −\mathrm{3x}−\mathrm{12y}+\mathrm{20} \\ $$

Question Number 191731 Answers: 1 Comments: 0

Use laws of algebra to prove the following (a)[(B−A)u(A−B)]=[(AuB)−(AnB)] (b)A▽(AnB)=A−B

$$\mathrm{Use}\:\mathrm{laws}\:\mathrm{of}\:\mathrm{algebra}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{the}\:\mathrm{following} \\ $$$$\left(\mathrm{a}\right)\left[\left(\mathrm{B}−\mathrm{A}\right)\mathrm{u}\left(\mathrm{A}−\mathrm{B}\right)\right]=\left[\left(\mathrm{AuB}\right)−\left(\mathrm{AnB}\right)\right] \\ $$$$\left(\mathrm{b}\right)\mathrm{A}\bigtriangledown\left(\mathrm{AnB}\right)=\mathrm{A}−\mathrm{B} \\ $$

Question Number 191723 Answers: 1 Comments: 0

∫_0 ^(2π) 3sintcost dt

$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mathrm{3}{sintcost}\:{dt} \\ $$

Question Number 191753 Answers: 1 Comments: 0

Question Number 191717 Answers: 0 Comments: 0

Question Number 191716 Answers: 0 Comments: 0

Question Number 191715 Answers: 1 Comments: 0

Question Number 191714 Answers: 1 Comments: 0

Question Number 191713 Answers: 0 Comments: 0

Question Number 191710 Answers: 0 Comments: 1

What is the remainder f 149! when divided by 139?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{f}\:\mathrm{149}!\:\mathrm{when}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\mathrm{139}? \\ $$

Question Number 191706 Answers: 2 Comments: 0

Question Number 191688 Answers: 1 Comments: 9

Divide a 113mm line into ratio 1:2:4

$$\mathrm{Divide}\:\mathrm{a}\:\mathrm{113mm}\:\mathrm{line}\:\mathrm{into}\:\mathrm{ratio} \\ $$$$\mathrm{1}:\mathrm{2}:\mathrm{4} \\ $$

Question Number 191680 Answers: 1 Comments: 0

Find the remainder of 67! when divided by 7!

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{of}\:\mathrm{67}!\:\mathrm{when}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\mathrm{7}! \\ $$

Question Number 191676 Answers: 2 Comments: 2

Question Number 191675 Answers: 2 Comments: 2

Solve for x : (x − (1/x))^(1/2) + (1 − (1/x))^(1/2) = x

$$\mathrm{Solve}\:\mathrm{for}\:{x}\:: \\ $$$$\left({x}\:−\:\frac{\mathrm{1}}{{x}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:+\:\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{{x}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:=\:{x} \\ $$

Question Number 191668 Answers: 0 Comments: 0

∫ ((√x)/( (√((1−x^2 )^3 ))))dx = ?

$$\:\:\:\:\int\:\:\frac{\sqrt{\boldsymbol{\mathrm{x}}}}{\:\sqrt{\left(\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)^{\mathrm{3}} }}\boldsymbol{\mathrm{dx}}\:\:\:\:=\:\:? \\ $$

Question Number 191663 Answers: 4 Comments: 0

2^a +4^b +8^c =328 find a,b and c when(a,b,c)is natual number

$$\mathrm{2}^{{a}} +\mathrm{4}^{{b}} +\mathrm{8}^{{c}} =\mathrm{328} \\ $$$${find}\:{a},{b}\:{and}\:{c} \\ $$$${when}\left({a},{b},{c}\right){is}\:{natual}\:{number} \\ $$

Question Number 191652 Answers: 1 Comments: 1

Question Number 191642 Answers: 0 Comments: 1

Comparer [OHDF] avec [ABHEF] (avec preuve) Sachant que: OH∣∣DF HE=EF OB=4OA CH=2BC et BC<OB.

$$\mathrm{Comparer}\:\left[\mathrm{OHDF}\right]\:\:\mathrm{avec}\:\left[\mathrm{ABHEF}\right] \\ $$$$\left({avec}\:{preuve}\right) \\ $$$$\:\:{Sa}\mathrm{c}{h}\mathrm{a}{nt}\:{que}: \\ $$$$\mathrm{OH}\mid\mid\mathrm{DF}\:\:\:\mathrm{HE}=\mathrm{EF}\:\:\:\mathrm{OB}=\mathrm{4OA} \\ $$$$\:\:\:\mathrm{CH}=\mathrm{2BC}\:\mathrm{et}\:\mathrm{BC}<\mathrm{OB}.\:\:\: \\ $$$$ \\ $$

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