(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.
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.
Show that two m×n matrices are equivalent if and only if they
have the same invariant factors, i.e. (by Problem 4), if and only
if they have the same Smith normal form.
Person A and B both have the same income, and there are only two
goods in the market: good X has the price of $6, good Y has the
price of $4. Considering A and B may have different bundles for
purchasing good X and Y, but they are on the same budget line with
the same MRT and MRS, will their consumer surplus always be the
same? or does it depend on the bundles?
This is the only info that we have:
2. It is illegal for any two firms that sell similar products to
engage in price fixing agreements. Violating the anti-trust laws
can bring both civil and criminal prosecutions. Nevertheless, price
fixing does take place. Examples would be found at the service
plazas along the NY State Thruway and the NJ Turnpike. Each
location has a small number of fast food restaurants. Each fast
food restaurant belongs to a different firm, which...
This is the only info that we have:
2. It is illegal for any two firms that sell similar products to
engage in price fixing agreements. Violating the anti-trust laws
can bring both civil and criminal prosecutions. Nevertheless, price
fixing does take place. Examples would be found at the service
plazas along the NY State Thruway and the NJ Turnpike. Each
location has a small number of fast food restaurants. Each fast
food restaurant belongs to a different firm, which...
Show that a set is convex if and only if its intersection with
any line is convex. Show that a set is affine if and only if its
intersection with any line is affine.