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AllQuestion and Answers: Page 210

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question about tinkutara how can an answer be placed in a box.

$$\:\:\underline{\mathrm{question}\:\mathrm{about}\:\mathrm{tinkutara}} \\ $$$$\:\:\mathrm{how}\:\mathrm{can}\:\mathrm{an}\:\mathrm{answer}\:\mathrm{be}\:\mathrm{placed}\:\: \\ $$$$\:\:\mathrm{in}\:\mathrm{a}\:\mathrm{box}. \\ $$

Question Number 193922 Answers: 4 Comments: 0

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Show that the kernel of a group homomorhism θ : G → H is a normal subgroup. Hint: Check the existence of the combination g^(−1) kg in the kernel.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{kernel}\:\mathrm{of}\:\mathrm{a}\:\mathrm{group}\:\mathrm{homomorhism} \\ $$$$\theta\::\:\mathrm{G}\:\rightarrow\:\mathrm{H}\:\mathrm{is}\:\mathrm{a}\:\mathrm{normal}\:\mathrm{subgroup}. \\ $$$$\mathrm{Hint}:\:\mathrm{Check}\:\mathrm{the}\:\mathrm{existence}\:\mathrm{of}\:\mathrm{the}\:\mathrm{combination} \\ $$$$\mathrm{g}^{−\mathrm{1}} \mathrm{kg}\:\mathrm{in}\:\mathrm{the}\:\mathrm{kernel}. \\ $$

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Ques. 12 If Y = {0, 1, 2, 3, 4} is transversal for 5Z in (Z, +). Show whether or not Y is a subgroup of 5Z subgroup under addition of integers modulo of 5

$$\mathrm{Ques}.\:\mathrm{12} \\ $$$$\mathrm{If}\:\mathrm{Y}\:=\:\left\{\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\right\}\:\mathrm{is}\:\mathrm{transversal}\:\mathrm{for}\:\mathrm{5}\mathbb{Z} \\ $$$$\mathrm{in}\:\left(\mathbb{Z},\:+\right).\:\mathrm{Show}\:\mathrm{whether}\:\mathrm{or}\:\mathrm{not}\:\mathrm{Y}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{subgroup}\:\mathrm{of}\:\mathrm{5}\mathbb{Z}\: \\ $$$$ \\ $$$$\mathrm{subgroup}\:\mathrm{under}\:\mathrm{addition}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{modulo} \\ $$$$\mathrm{of}\:\mathrm{5} \\ $$

Question Number 193893 Answers: 1 Comments: 0

Ques. 11 Let {H_α } ∈ Ω be a family of subgroup of a group G then prove that ∩_(α=Ω) H_α is also a subgroup Ques. 12 Using GAP, find the elements A, B and C in D_5 such that AB = BC but A ≠ C.

$$\mathrm{Ques}.\:\mathrm{11} \\ $$$$\:\:\:\:\:\mathrm{Let}\:\left\{\mathrm{H}_{\alpha} \right\}\:\in\:\Omega\:\mathrm{be}\:\mathrm{a}\:\mathrm{family}\:\mathrm{of}\:\mathrm{subgroup}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{group}\:\mathrm{G}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\underset{\alpha=\Omega} {\cap}\mathrm{H}_{\alpha} \:\mathrm{is}\:\mathrm{also}\:\mathrm{a} \\ $$$$\mathrm{subgroup} \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{12}\: \\ $$$$\:\:\:\:\:\mathrm{Using}\:\mathrm{GAP},\:\mathrm{find}\:\mathrm{the}\:\mathrm{elements}\:\mathrm{A},\:\mathrm{B}\:\mathrm{and}\: \\ $$$$\mathrm{C}\:\mathrm{in}\:\mathrm{D}_{\mathrm{5}} \:\mathrm{such}\:\mathrm{that}\:\mathrm{AB}\:=\:\mathrm{BC}\:\mathrm{but}\:\mathrm{A}\:\neq\:\mathrm{C}. \\ $$

Question Number 193892 Answers: 2 Comments: 0

Ques. Find the number of integers in the set S={1,2,3,...,60} which are not divisible by 2 nor by 3 nor by 5. Hello

$$\mathrm{Ques}. \\ $$$$\:\:\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integers}\:\mathrm{in}\:\mathrm{the}\:\mathrm{set} \\ $$$$\mathrm{S}=\left\{\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{60}\right\}\:\mathrm{which}\:\mathrm{are}\:\mathrm{not}\:\mathrm{divisible} \\ $$$$\mathrm{by}\:\mathrm{2}\:\mathrm{nor}\:\mathrm{by}\:\mathrm{3}\:\mathrm{nor}\:\mathrm{by}\:\mathrm{5}. \\ $$$$ \\ $$$$\mathrm{Hello} \\ $$

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If a 9000kg water flows in a minute through a pipe of cross sectional area 0.3m², what is the speed of water in the pipe?

$$ \\ $$If a 9000kg water flows in a minute through a pipe of cross sectional area 0.3m², what is the speed of water in the pipe?

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a,b,c,d,e,f, are + real numbers prove: (a/(b+c))+(b/(c+d))+(c/(d+e))+(d/(e+f))+(e/(f+a))+(f/(a+b))≥3

$${a},{b},{c},{d},{e},{f},\:{are}\:+\:{real}\:{numbers} \\ $$$${prove}: \\ $$$$\frac{{a}}{{b}+{c}}+\frac{{b}}{{c}+{d}}+\frac{{c}}{{d}+{e}}+\frac{{d}}{{e}+{f}}+\frac{{e}}{{f}+{a}}+\frac{{f}}{{a}+{b}}\geqslant\mathrm{3} \\ $$

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Ques. 8 Find the signum (sign or sgn) of the permutation θ=(12345678). Hint : for any permutation β, take sgn β = {_(−1 if β is odd) ^(1 if β is even) Ques. 9 Prove that ∣S_n ∣ = n!. Ques. 10 Provd that for b∈S_n , sgn b = sgn b^(−1) .

$$\mathrm{Ques}.\:\mathrm{8}\: \\ $$$$\:\:\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{signum}\:\left(\mathrm{sign}\:\mathrm{or}\:\mathrm{sgn}\right)\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{permutation}\:\theta=\left(\mathrm{12345678}\right). \\ $$$$\mathrm{Hint}\::\:\mathrm{for}\:\mathrm{any}\:\mathrm{permutation}\:\beta,\:\mathrm{take} \\ $$$$\mathrm{sgn}\:\beta\:=\:\left\{_{−\mathrm{1}\:\:\:\:\:\:\:\mathrm{if}\:\beta\:\mathrm{is}\:\mathrm{odd}} ^{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:\beta\:\mathrm{is}\:\mathrm{even}} \right. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{9} \\ $$$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}\:\mid\mathrm{S}_{\mathrm{n}} \mid\:=\:\mathrm{n}!. \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{10} \\ $$$$\:\:\:\:\:\:\mathrm{Provd}\:\mathrm{that}\:\mathrm{for}\:\mathrm{b}\in\mathrm{S}_{\mathrm{n}} ,\:\mathrm{sgn}\:\mathrm{b}\:=\:\mathrm{sgn}\:\mathrm{b}^{−\mathrm{1}} \:. \\ $$

Question Number 193866 Answers: 1 Comments: 0

prove Σ_(i=1) ^(+∞) (1/n^i )=(1/(n−1)) n∈N^∗ and if n>0∧ n∈R is it right?

$${prove} \\ $$$$\:\:\underset{{i}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{i}} }=\frac{\mathrm{1}}{{n}−\mathrm{1}}\:\:\:\:\:\:{n}\in\mathbb{N}^{\ast} \\ $$$${and}\:{if}\:{n}>\mathrm{0}\wedge\:{n}\in\mathbb{R} \\ $$$${is}\:{it}\:{right}? \\ $$

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lim_(x→0) ((1−(1/2)x^2 −cos ((x/(1−x^2 ))))/x^4 ) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}^{\mathrm{2}} −\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\right)}{\mathrm{x}^{\mathrm{4}} }\:=? \\ $$

Question Number 193853 Answers: 0 Comments: 0

Let n & k be positive integers and let S be a set of n points in The plane such that : For any point P of S there are at least K points of S Equidistant from p Prove that k<(1/2)+(√(2n))

$$ \\ $$$$\boldsymbol{{Let}}\:\boldsymbol{{n}}\:\&\:\boldsymbol{{k}}\:\boldsymbol{{be}}\:\boldsymbol{{positive}}\:\boldsymbol{{integers}}\:\boldsymbol{{and}}\:\boldsymbol{{let}} \\ $$$$\boldsymbol{{S}}\:\boldsymbol{{be}}\:\boldsymbol{{a}}\:\boldsymbol{{set}}\:\boldsymbol{{of}}\:\boldsymbol{{n}}\:\boldsymbol{{points}}\:\boldsymbol{{in}}\:\boldsymbol{{The}}\:\boldsymbol{{plane}}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:: \\ $$$$\boldsymbol{{For}}\:\boldsymbol{{any}}\:\boldsymbol{{point}}\:\boldsymbol{{P}}\:\boldsymbol{{of}}\:\boldsymbol{{S}}\:\boldsymbol{{there}}\:\boldsymbol{{are}}\:\boldsymbol{{at}}\:\boldsymbol{{least}}\:\boldsymbol{{K}}\:\boldsymbol{{points}}\:\boldsymbol{{of}}\:\boldsymbol{{S}}\:\boldsymbol{{Equidistant}}\:\boldsymbol{{from}}\:\boldsymbol{{p}} \\ $$$$\boldsymbol{{Prove}}\:\boldsymbol{{that}}\:\boldsymbol{{k}}<\frac{\mathrm{1}}{\mathrm{2}}+\sqrt{\mathrm{2}\boldsymbol{{n}}} \\ $$

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