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Set TheoryQuestion and Answers: Page 4

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 ...?

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

Question Number 127165 Answers: 0 Comments: 0

Question Number 126808 Answers: 2 Comments: 0

Question Number 126765 Answers: 1 Comments: 0

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

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

Question Number 124899 Answers: 0 Comments: 0

$$ \\ $$

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.

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

Question Number 120702 Answers: 0 Comments: 0

Question Number 120044 Answers: 2 Comments: 0

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

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

Question Number 117597 Answers: 0 Comments: 1

Let A, B, and C be three sets and X be the set of all elements which belong to exactly two of the sets A,B and C. Prove that X is equal to (A∪B∪C)−[AΔ(BΔC)]

$$\mathrm{Let}\:\mathrm{A},\:\mathrm{B},\:\mathrm{and}\:\mathrm{C}\:\mathrm{be}\:\mathrm{three}\:\mathrm{sets}\:\mathrm{and}\:\mathrm{X}\:\mathrm{be}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all} \\ $$$$\mathrm{elements}\:\mathrm{which}\:\mathrm{belong}\:\mathrm{to}\:\mathrm{exactly}\:\mathrm{two}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sets}\:\mathrm{A},\mathrm{B} \\ $$$$\mathrm{and}\:\mathrm{C}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{X}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{A}\cup\mathrm{B}\cup\mathrm{C}\right)−\left[\mathrm{A}\Delta\left(\mathrm{B}\Delta\mathrm{C}\right)\right] \\ $$

Question Number 114236 Answers: 2 Comments: 0

70% of the employees in a multinational corporation have VCD players, 75% have microwave ovens, 80% have ACs and 85% have washing machines. At least what percentage of employees has all four gadgets?

$$\mathrm{70\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{employees}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{multinational}\:\mathrm{corporation}\:\mathrm{have}\:\mathrm{VCD} \\ $$$$\mathrm{players},\:\mathrm{75\%}\:\mathrm{have}\:\mathrm{microwave}\:\mathrm{ovens}, \\ $$$$\mathrm{80\%}\:\mathrm{have}\:\mathrm{ACs}\:\mathrm{and}\:\mathrm{85\%}\:\mathrm{have}\:\mathrm{washing} \\ $$$$\mathrm{machines}.\:\mathrm{At}\:\mathrm{least}\:\mathrm{what}\:\mathrm{percentage}\:\mathrm{of} \\ $$$$\mathrm{employees}\:\mathrm{has}\:\mathrm{all}\:\mathrm{four}\:\mathrm{gadgets}? \\ $$

Question Number 114235 Answers: 2 Comments: 0

Let A={(n,2n):n∈N} and B={(2n,3n):n∈N}. Then A∩B is equal to

$$\mathrm{Let}\:\mathrm{A}=\left\{\left(\mathrm{n},\mathrm{2n}\right):\mathrm{n}\in\mathrm{N}\right\}\:\mathrm{and} \\ $$$$\mathrm{B}=\left\{\left(\mathrm{2n},\mathrm{3n}\right):\mathrm{n}\in\mathrm{N}\right\}.\:\mathrm{Then}\:\mathrm{A}\cap\mathrm{B}\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 114233 Answers: 1 Comments: 0

If two sets A and B are having 99 elements in common, then the number of elements common to each of the sets A×B and B×A is

$$\mathrm{If}\:\mathrm{two}\:\mathrm{sets}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{having}\:\mathrm{99} \\ $$$$\mathrm{elements}\:\mathrm{in}\:\mathrm{common},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{elements}\:\mathrm{common}\:\mathrm{to}\:\mathrm{each} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{sets}\:\mathrm{A}×\mathrm{B}\:\mathrm{and}\:\mathrm{B}×\mathrm{A}\:\mathrm{is} \\ $$

Question Number 114061 Answers: 1 Comments: 0

Question Number 104694 Answers: 0 Comments: 0

Question Number 104544 Answers: 2 Comments: 0

Given a_(n+1) = 5a_n −6a_(n−1) .If a_1 = 10 and a_2 = 26, find a_n ?

$$\mathcal{G}{iven}\:{a}_{{n}+\mathrm{1}} \:=\:\mathrm{5}{a}_{{n}} −\mathrm{6}{a}_{{n}−\mathrm{1}} \\ $$$$.\mathcal{I}{f}\:{a}_{\mathrm{1}} =\:\mathrm{10}\:{and}\:{a}_{\mathrm{2}} =\:\mathrm{26},\:{find}\:{a}_{{n}} ? \\ $$

Question Number 102357 Answers: 2 Comments: 0

for A={1,2,3,4,5,6,7},compute the number of: (a) Subsets of A. (b) Nonempty subsets of A. (c) proper subsets of A. (d) Non empty proper subset of A. (e) Subsets of A containing three element. (f) Subsets of A containing 1,2. (g) Proper subsets of A containing 1,2. (h) Subset of A with an even number of element. (i) Subset of A with an odd number of element. (j) Subsets of A with an odd number of elements, including the element 3.

$${for}\:{A}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{7}\right\},{compute}\:{the}\:{number}\:{of}: \\ $$$$\left({a}\right)\:{Subsets}\:{of}\:{A}. \\ $$$$\left({b}\right)\:{Nonempty}\:{subsets}\:{of}\:{A}. \\ $$$$\left({c}\right)\:{proper}\:{subsets}\:{of}\:{A}. \\ $$$$\left({d}\right)\:{Non}\:{empty}\:{proper}\:{subset}\:{of}\:{A}. \\ $$$$\left({e}\right)\:{Subsets}\:{of}\:{A}\:{containing}\:{three}\:{element}. \\ $$$$\left({f}\right)\:{Subsets}\:{of}\:{A}\:{containing}\:\mathrm{1},\mathrm{2}. \\ $$$$\left({g}\right)\:{Proper}\:{subsets}\:{of}\:{A}\:{containing}\:\mathrm{1},\mathrm{2}. \\ $$$$\left({h}\right)\:{Subset}\:{of}\:{A}\:{with}\:{an}\:{even}\:{number}\:{of}\:{element}. \\ $$$$\left({i}\right)\:{Subset}\:{of}\:{A}\:{with}\:{an}\:{odd}\:{number}\:{of}\:{element}. \\ $$$$\left({j}\right)\:{Subsets}\:{of}\:{A}\:{with}\:{an}\:{odd}\:{number}\:{of}\:{elements},\:{including}\:{the}\:{element}\:\mathrm{3}. \\ $$

Question Number 101891 Answers: 3 Comments: 0

if x a integer number , when divided 8 has remainder 5 and divided 5 has remainder 2. find x

$${if}\:{x}\:{a}\:{integer}\:{number}\:,\:{when}\:{divided}\:\mathrm{8} \\ $$$${has}\:{remainder}\:\mathrm{5}\:{and}\:{divided}\:\mathrm{5}\:{has}\:{remainder} \\ $$$$\mathrm{2}.\:{find}\:{x} \\ $$

Question Number 98806 Answers: 0 Comments: 1

for a is integer number such that ∣∣x−1∣ −2∣ ≤ a exactly has 2013 solution

$$\mathrm{for}\:{a}\:\mathrm{is}\:\mathrm{integer}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mid\mid{x}−\mathrm{1}\mid\:−\mathrm{2}\mid\:\leqslant\:{a}\:\:\mathrm{exactly}\:\mathrm{has}\:\mathrm{2013} \\ $$$$\mathrm{solution} \\ $$

Question Number 97438 Answers: 1 Comments: 1

If −3≤x≤4, −2≤y≤5, 4≤z≤10 , find the greatest value of w = z−xy

$$\mathrm{If}\:−\mathrm{3}\leqslant\mathrm{x}\leqslant\mathrm{4},\:−\mathrm{2}\leqslant\mathrm{y}\leqslant\mathrm{5},\:\mathrm{4}\leqslant\mathrm{z}\leqslant\mathrm{10} \\ $$$$,\:\mathrm{find}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{w}\:=\:\mathrm{z}−\mathrm{xy}\: \\ $$

Question Number 93950 Answers: 2 Comments: 0

Let ∗ be the binary operation on N given by a∗b=L.C.M. of a and b. Find (i) 5∗7 , 20∗16 (ii) is ∗ communitative? (iii) is ∗ associative? (iv)Find the identity of ∗ in N (v) which elements of N are invertible for the operation ∗?

$${Let}\:\ast\:{be}\:{the}\:{binary}\:{operation}\:{on}\:\mathrm{N} \\ $$$${given}\:{by}\:\mathrm{a}\ast\mathrm{b}=\mathrm{L}.\mathrm{C}.\mathrm{M}.\:{of}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{\mathrm{b}}.\:\mathrm{F}{ind} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{5}\ast\mathrm{7}\:,\:\:\mathrm{20}\ast\mathrm{16}\:\:\:\left(\mathrm{ii}\right)\:\mathrm{is}\:\ast\:{communitative}? \\ $$$$\left(\mathrm{iii}\right)\:\mathrm{is}\:\ast\:{associative}? \\ $$$$\left(\mathrm{iv}\right)\mathrm{F}{ind}\:{the}\:{identity}\:{of}\:\ast\:{in}\:\boldsymbol{\mathrm{N}} \\ $$$$\left(\mathrm{v}\right)\:\mathrm{which}\:\mathrm{elements}\:\mathrm{of}\:\boldsymbol{\mathrm{N}}\:{are}\:{invertible} \\ $$$$\:\:\:\:\:\:\:{for}\:{the}\:{operation}\:\ast? \\ $$

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