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 any two permutations σ,τ ∈ Sn have the same cycle structure if and only...

Show that any two permutations σ,τ ∈ Sn have the same cycle structure if and only if there exists γ ∈ Sn such that σ = γτγ^−1.

An inversion in σ ∈ Sn is a pair (i, j) such that i < j...

An inversion in σ ∈ Sn is a pair (i, j) such that i < j and σ(i) > σ(j). Take σ, τ ∈ Sn and take i < j. - Suppose (i, j) is not an inversion in τ. Show that (i, j) is an inversion in στ if and only if (τ(i), τ(j)) is an inversion in σ. - Suppose (i, j) is an inversion in τ. Show that (i, j) is an inversion in στ if and...

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?

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