Consider the following permutations in S7:2 3 4 5 6 74 5 62 3 5 6 1 7 411and т —2 37O =3712 4 6(a) Compute the following products: (a) oT (b) TO (c) t²o (d) o-1 (e) o-170.(b) Write the following as products of disjoint cycles: (a) o (b) t (c) oT (d) o-1 (e)-0.

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Answered: Consider the following permutations in…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please help to solve this. Does this question use combinatorics? Since we are told that marbles are selected at random (together, without replacement). A box has 1 red and 3 blue marbles.Two marble are selected at random (together, without replacement).Let R be the number of reds we get.Let B be the number of blues we get.Compute Cov(R, B).
Suppose a permutation s swaps the ﬁrst two elements of {1, 2, ..., n}. A permutation r rotates the elements: r(1)=2, r(2)=3, ... r(n-1)=n, r(n)=1. We can thus write in cycle notation: s=(1 2), r=(1 2 3 4 ... n). What does the permutation rm sr -m do?
Part g please
QUESTION 1 By using the laws of indices, the expression 2x 2x simplifies to: O 3x8 O 873 O 37 8 O 87 3 O 37 6 O 3r24 O 8r 6 QUESTION 2 By expanding out the brackets and collecting like terms, the expression (VE – V2) (vE +4/2) simplifies to: O r + 8VT + 3/2 O x + 3/2x – 8 O 3x + V2x +8 O v2x + 3/T – 8
Conjugate elements. Let 0,T E Sn. Define d = TOT. (o and d are called conjugate elements; see Chapter 6.) (a) Show that if o (i) = j, then 8(r (i)) = T(j). (b) Explain that the previous part says that if you apply T to each of the entries in the cycle notation of o, then you get the cycle notation for 8. In other words, if o has cycle decomposition (a1 a2 aki k1)(bị b2 bk2) · ·· . .. ... 3. The Alternating Groups Then d has cycle decomposition (7(a1) T(a2) ·… (c) Illustrate the previous part by letting o = (1 6 3)(7 5 2), and quickly writing down the cycle decomposition for TOT. Check your answer by actually finding the product rOT. T(bk2)) .... (1 4 3 8)(2 6 5), т — T(ak,))(t(b1) T(b2) .. ..

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Consider the following permutations in S7: 1 2 3 4 5 6 7 1 2 3 4 5 6 7 3 5 7 1 = and t = 2 4 6 2 3 5 6 1 7 4 ( a) Compute the following products: (a) στ (b ) Το ( c) Τσ (d) στ1 (e) σ1Τσ (b) Write the following as products of disjoint cycles: (a) o (b) T (c) OT (d) o-1 (e) o-l70.