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1. Let A be a finite set with subsets A1, ..., An, and let d1, ..., dn ∈ N. Show that there are disjoint subsets Dk ⊆ Ak, with |Dk| = dk for all k ≤ n, if and only if | S i∈I Ai | ≥ P i∈I di for all I ⊆ {1, ..., n}.
Hint: Construct a bipartite graph in which A is one side, and the other side consists of a suitable number of copies of the sets Ai . Define the edge set of the graph so that the desired result can be derived from the marriage theorem.
2. Find a bipartite graph with a set of preferences such that no matching of maximum size is stable and no stable matching has maximum size. Find a non-bipartite graph with a set of preferences that has no stable matching.
Hint: Try C 6 . Change occurs most likely if unhappy vertices can bring it about without having to ask the happy ones. (If philosophy does not help, try K3 .)
suppose that a connected planar simple graph with edges and vertices contain no simple circuit of length 4 or less.
7 people from town B, and 5 people from town C. If a council consisting of 5 people is randomly selected, find the probability that 3 are from town A and 2 are from town B. How would I show my work on this equation?
Math: Linear Transformation of the Space of Polynomials, Which of the following is a linear transformation T of the space of polynomials? Circle the letters corresponding to correct answers.
you win 200000 in state lotery. you will get paid 100000 at the end of each year for the next 20 years. how much have
a certain type of light bulb has a life time can be represented by a normal distribution with a mean of 500 hours and a standard deviation of 100 hours
Please explain how to find the equations of the tangent line and the normal line to the graph of the equation at the indicated point and achieve the specified answer.
What would your decision be in this case using a 5% significance level?
A farmer wants to set aside a rectangular plot of land to contain 580 square meters. If the width is 19 less than the length, find the dimensions of the plot.
Where t is the number of years after January 1, 1998 and C(t) is the number of conncections in millions.
The formula d = , where h equals the height, in feet, of the viewer's eyes, estimates the distance d, in miles, to the horizon from the viewer. Find the distance to the horizon for a person who eyes are 6 ft above the ground.
In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor a particular candidate.
(a) Prove that a closed k-form w on S^k is exact if and only if int_(S^k) w = 0. [Hint: dim H^k(S^k) = 1.](b) Conclude that the linear map int_(S^k): H^k(S^k) -> R is an isomorphism.
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