Show that for all σ ∈ Sn we have sgn(σ) = sgn(σ−1). Does σ = (1,2,3,5,4)−1...

Show that for all σ ∈ Sn we have sgn(σ) = sgn(σ−1). Does σ = (1,2,3,5,4)−1 ∈ S13 belong to the alternating group A13? Justify your answer

Expert Solution

Note that, transposition means cycle of length 2.

Colby Messinger answered 2 years ago

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all...

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many sn belong to [a,b], then lim sn belongs to [a, b].

Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>

Show that a dilation σ by a factor of r scales all distances by a factor...

Show that a dilation σ by a factor of r scales all distances by a factor of r. That is, for all points A and B, we have: |σ(A)σ(B)| = r|AB|

3. Every element of Sn can be written as a product of disjoint cycles. If σ...

3. Every element of Sn can be written as a product of disjoint cycles. If σ = (i1 i2)(j1 j2)(k1 k2 k3 k4), with the cycles disjoint, we say that σ has cyclic structure ( )( )( ). (a) Find all possible cyclic structures in S7. Hint: There are 15. (b) Using part (a), find all possible orders in S7. (c) Find all possible orders in A7.

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn...

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn where k ≤ n. (a) For σ ∈ Sn, prove that στσ-1 = (σ(1), σ(2), . . . , σ(k)). (b) Let ρ be any cycle of length k in Sn. Prove that there exists an element σ ∈ Sn so that στσ-1 = ρ.

Consider the sequence sn defined as: s0 = 1 s1 = 1 sn = 2sn-1 +...

Consider the sequence sn defined as: s0 = 1 s1 = 1 sn = 2sn-1 + sn-2 What is the base case for this recursive relation? Find s5 Write the pseudocode for a recursive function to find Sn for any arbitrary value of n. Create a non-recursive formula for finding the nth term in the sequence in O(1) time.

Let Σ ⊆ P rop(A). Show that Σ|− p iff Σ ∪ {¬p} is unsatisfifiable.

You have a normal population of scores with μ = 50 and σ = 14. We...

You have a normal population of scores with μ = 50 and σ = 14. We obtain a random sample of n = 36. What is the probability that the sample mean will be less than 54?

We have a normal probability distribution for variable x with μ = 24.8 and σ =...

We have a normal probability distribution for variable x with μ = 24.8 and σ = 3.7. For each of the following, make a pencil sketch of the distribution & your finding. Using 5 decimals, find: (a)[1] PDF(x = 30); [use NORM.DIST(•,•,•,0)] (b)[1] P(x ≤ 27); [use NORM.DIST(•,•,•,1)] (c)[1] P(x ≥ 21); [elaborate, then use NORM.DIST(•,•,•,1)] (d)[2] P(22 ≤ x ≤ 26); [elaborate, then use NORM.DIST(•,•,•,1)] (e)[3] P(20 ≤ x ≤ 28); [z‐scores, then use NORM.S.DIST(•,1)] (f)[1] x* so that...

(The “conjugation rewrite lemma”.) Let σ and τ be permutations. (a) Show that if σ maps...

(The “conjugation rewrite lemma”.) Let σ and τ be permutations. (a) Show that if σ maps x to y then στ maps τ(x) to τ(y). (b) Suppose that σ is a product of disjoint cycles. Show that στ has the same cycle structure as σ; indeed, wherever (... x y ...) occurs in σ, (... τ(x) τ(y) ...) occurs in στ.

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