99年私立元智大學生醫所離散數學線上測驗

1.: How many ways are there to distribute r distinct objects into five (distinct) boxes with at least one empty box?

2.: G is a graph. Prove that if G is acyclic and V_G = E_G + 1, then G is a tree. Note that E_G is the number of the vertices of G, and E_G is the number of the edges of G.

3.: Given a simple graph G, it is nature to define the line graph, L(G), where L(G) contains a vertex for each edge of G and an edge between two vertices if and only if the corresponding edges of G share a vertex. Show us some examples to explain that if G is Eulerian, then L(G) is Eulerian and Hamiltonian.

5.: How many r-digit quaternary sequences (sequences using only the digits 0, 1, 2, 3) have at least one 1, one 2, and one 3?

6.: Consider the following relations, please answer the questions of which relations are symmetric, which are antisymmetric, which are reflexive, which are irreflexive, and which are transitive. Please give us their closures if they have:
(a) IsMarriedTo
(b) IsParentOf
(c) IsEqualTo
(d) IsPartOf
(e) HigherValue
(f) IsAncestorOf
(g) ApproximatelyEqual (if |x − y| < ε, where ε is a very small given value, we say that x and y are ApproximatelyEqual)

購買題庫後，可使用那些功能？