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Set TheoryQuestion and Answers: Page 3

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find two irrational numbers between 0.333.... and 0.444...

$$\mathrm{find}\:\mathrm{two}\:\mathrm{irrational}\:\mathrm{numbers}\: \\ $$$$\mathrm{between}\:\mathrm{0}.\mathrm{333}....\:\mathrm{and}\:\mathrm{0}.\mathrm{444}... \\ $$$$ \\ $$

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Prove the set {1,2,3,...,1989} can be expressed as the disjoint union of A_1 ,A_2 ,...,A_(117) such that (i) each A_i contains the same number of elements ,and (ii) the sum of all elements of each A_i is the same for i=1,2,3,...,m

$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{set}\:\left\{\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{1989}\right\} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{as}\:\mathrm{the}\:\mathrm{disjoint} \\ $$$$\mathrm{union}\:\mathrm{of}\:\mathrm{A}_{\mathrm{1}} ,\mathrm{A}_{\mathrm{2}} ,...,\mathrm{A}_{\mathrm{117}} \:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{each}\:\mathrm{A}_{\mathrm{i}} \:\mathrm{contains}\:\mathrm{the}\:\mathrm{same}\:\mathrm{number}\:\mathrm{of}\:\mathrm{elements}\:,\mathrm{and} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{each}\:\mathrm{A}_{\mathrm{i}} \:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{for}\:\mathrm{i}=\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{m} \\ $$

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Question Number 129634 Answers: 1 Comments: 0

sin^ x+cos^ x=1 proof by step by step or by showing all steps b/c it is my assignment

$$\boldsymbol{{sin}}^{ } \boldsymbol{{x}}+\boldsymbol{{cos}}^{ } \boldsymbol{{x}}=\mathrm{1}\:\mathrm{proof}\:\mathrm{by}\:\mathrm{step}\:\mathrm{by}\:\mathrm{step}\:\mathrm{or}\:\mathrm{by}\:\mathrm{showing}\:\mathrm{all}\:\mathrm{steps}\: \\ $$$$\mathrm{b}/\mathrm{c}\:\mathrm{it}\:\mathrm{is}\:\mathrm{my}\:\mathrm{assignment} \\ $$

Question Number 128246 Answers: 0 Comments: 0

...nice calculus... suppose that :: m=((4^p −1)/3) , where p is a prime number and p>3. prove that ::: 2^(m−1) ≡^m 1 ...?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:{suppose}\:{that}\:\:::\:{m}=\frac{\mathrm{4}^{{p}} −\mathrm{1}}{\mathrm{3}}\:,\:\:{where} \\ $$$$\:\:\:\:\:\:\:{p}\:\:{is}\:\:{a}\:{prime}\:{number}\:{and}\:\:{p}>\mathrm{3}. \\ $$$${prove}\:\:{that}\:\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{{m}−\mathrm{1}} \:\overset{{m}} {\equiv}\:\mathrm{1}\:\:\:\:...? \\ $$$$\: \\ $$

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...elementary mathematics... if 13 ∣9^(51) +k+1 , k∈N then k_((min)) =?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{elementary}\:\:{mathematics}... \\ $$$$\:\:\:{if}\:\:\:\:\:\mathrm{13}\:\mid\mathrm{9}^{\mathrm{51}} +{k}+\mathrm{1}\:\:\:,\:{k}\in\mathbb{N} \\ $$$$\:\:\:\:\:\:\:\:{then}\:\:\:{k}_{\left({min}\right)} \:=? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$

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Question Number 124409 Answers: 1 Comments: 0

write A′∪ B′ in a disjoint set

$${write}\:\mathrm{A}'\cup\:{B}'\:{in}\:{a}\:{disjoint}\:{set} \\ $$

Question Number 123287 Answers: 1 Comments: 0

... nice calculus ... number theory prove thar ::: 2^(32) +1≡^(641) 0 ✓ notice: without calculator and only with the use of congruence properties..

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}\:... \\ $$$$\:\:\:\:\:\:\:{number}\:{theory} \\ $$$$\:\:\:\:\:\:\:\:\:\:{prove}\:{thar}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{32}} +\mathrm{1}\overset{\mathrm{641}} {\equiv}\mathrm{0}\:\checkmark \\ $$$$\:\:\:{notice}:\:{without}\:{calculator}\:{and}\:{only} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{with}\:{the}\:{use}\:{of}\:{congruence}\:{properties}.. \\ $$

Question Number 122436 Answers: 0 Comments: 0

Mean : It is found by adding all the values of the observation and dividing it by the total number of observations. It is denoted by x^ . So, x^ = ((Σ_(i=1) ^n x_i )/n). For an ungrouped frequency distribution, it is x^ = ((Σ_(i = 1) ^n f_i x_i )/(Σ_(i = 1) ^n f_i )) .

$$\boldsymbol{\mathrm{Mean}}\::\:\mathrm{It}\:\mathrm{is}\:\mathrm{found}\:\mathrm{by}\:\mathrm{adding}\:\mathrm{all}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{the}\:\mathrm{observation}\:\mathrm{and}\:\mathrm{dividing}\:\mathrm{it}\:\mathrm{by}\:\mathrm{the} \\ $$$$\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{observations}.\:\mathrm{It}\:\mathrm{is}\:\mathrm{denoted}\:\mathrm{by}\:\bar {{x}}. \\ $$$$\mathrm{So},\:\bar {{x}}\:=\:\frac{\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{x}_{{i}} }{{n}}.\:\mathrm{For}\:\mathrm{an}\:\boldsymbol{\mathrm{ungrouped}}\:\boldsymbol{\mathrm{frequency}}\:\boldsymbol{\mathrm{distribution}},\:\mathrm{it}\:\mathrm{is}\:\bar {\boldsymbol{{x}}}\:=\:\frac{\underset{\boldsymbol{{i}}\:=\:\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\boldsymbol{{f}}_{\boldsymbol{{i}}} \boldsymbol{{x}}_{\boldsymbol{{i}}} }{\underset{\boldsymbol{{i}}\:=\:\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\boldsymbol{{f}}_{\boldsymbol{{i}}} }\:. \\ $$

Question Number 122426 Answers: 0 Comments: 0

In an exam 36% of people failed in physics and 49% failed in maths and 15% failed in both subject. If the total number of student that passed physics only is 680 find the total number of students that appeared in the exam .

$$ \\ $$$$\mathrm{In}\:\mathrm{an}\:\mathrm{exam}\:\mathrm{36\%}\:\mathrm{of}\:\mathrm{people}\:\mathrm{failed}\:\mathrm{in}\: \\ $$$$\mathrm{physics}\:\mathrm{and}\:\mathrm{49\%}\:\mathrm{failed}\:\mathrm{in}\:\mathrm{maths}\:\mathrm{and}\:\mathrm{15\%} \\ $$$$\mathrm{failed}\:\mathrm{in}\:\mathrm{both}\:\mathrm{subject}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{total}\: \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{student}\:\mathrm{that}\:\mathrm{passed}\:\mathrm{physics} \\ $$$$\mathrm{only}\:\mathrm{is}\:\mathrm{680}\:\mathrm{find}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{students}\:\mathrm{that}\:\mathrm{appeared}\:\mathrm{in}\:\mathrm{the}\:\mathrm{exam}\:. \\ $$

Question Number 121870 Answers: 1 Comments: 0

number theory: m,n ∈ N , (m,n)=1 prove : m^(ϕ(n)) +n^(ϕ(m)) ≡^(mn) 1 ϕ(n)=∣{x∈N∣ x<n , (x,n)=1}∣ .m.n.

$$\:\:\:\:\:\:{number}\:\:{theory}: \\ $$$$\:\:\:\:\:\:{m},{n}\:\in\:\mathbb{N}\:\:,\:\:\left({m},{n}\right)=\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\::\:\:\:{m}^{\varphi\left({n}\right)} +{n}^{\varphi\left({m}\right)} \overset{{mn}} {\equiv}\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\varphi\left({n}\right)=\mid\left\{{x}\in\mathbb{N}\mid\:{x}<{n}\:,\:\left({x},{n}\right)=\mathrm{1}\right\}\mid \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:.{m}.{n}. \\ $$

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Given a_(n+1) = ((2a_n )/((2n+1)(2n+2))) find a_n .

$${Given}\:{a}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}{a}_{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{2}\right)} \\ $$$${find}\:{a}_{{n}} . \\ $$

Question Number 119487 Answers: 2 Comments: 0

find element of set S = { ((x^3 −3x^2 +2)/(2x+1)) ∈ Z for x∈Z }

$${find}\:{element}\:{of}\:{set}\:{S}\:=\:\left\{\:\frac{{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{2}{x}+\mathrm{1}}\:\in\:\mathbb{Z}\:{for}\:{x}\in\mathbb{Z}\:\right\} \\ $$

Question Number 119401 Answers: 2 Comments: 0

let d be an application d:R^2 →R_+ d(x,y)=ln(1+((∣x−y∣)/(1+∣x−y∣))) shown that d is a distance on R^2 please help ★especially on triangular inequality

$$\boldsymbol{{let}}\:\boldsymbol{{d}}\:\boldsymbol{{be}}\:\boldsymbol{{an}}\:\boldsymbol{{application}} \\ $$$$\boldsymbol{{d}}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}_{+} \\ $$$$\boldsymbol{{d}}\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)=\boldsymbol{{ln}}\left(\mathrm{1}+\frac{\mid\boldsymbol{{x}}−\boldsymbol{{y}}\mid}{\mathrm{1}+\mid\boldsymbol{{x}}−\boldsymbol{{y}}\mid}\right) \\ $$$$\boldsymbol{{shown}}\:\boldsymbol{{that}}\:\boldsymbol{{d}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{distance}} \\ $$$$\boldsymbol{{on}}\:\mathbb{R}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{please}}\:\boldsymbol{{help}}\: \\ $$$$\:\bigstar\boldsymbol{{especially}}\:\boldsymbol{{on}}\:\boldsymbol{{triangular}} \\ $$$$\boldsymbol{{inequality}} \\ $$

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