Q1: 20+10 points Q2: 30 points Q3: 30 points Q4: 30 points Total: 100+10 points
Here are definitions of some terms, sets, predicates, and statements used on this exam.
Remember that a natural is a non-negative integer, so that the set N or all naturals is {0, 1, 2, 3,...}.
The Fibonacci numbers are a sequence of naturals defined by the rules F(0) = 0, F(1) = 1, and, for all n with n ≥ 1, F(n+1) = F(n) + F(n-1).
For every natural n, we define two graphs, a directed graph Dn and an undirected graph Un. Each has vertex set {0, 1,..., n}. The directed edges of Dn are (1) (i, i+1) for every i with 0 ≤ i ≤ n-1, (2) (i+2, i) for every even i with 0 ≤ i ≤ n-2, and (3) (i, i+2) for every odd with 1 ≤ i ≤ n-2.
We also view Dn as a labeled graph where each edge of type (1) above has cost 2, and the other edges have cost 3.
The undirected graph Un has the same vertex set as Dn, and an undirected edge whenever Dn has a directed edge. We will not have occasion to use edge costs in Un.
Directed graph D_5 with edge costs. The graph U_5 has the
same nodes and edges but all the edges are undirected.
___________ _____________
/ 3 \ / 3 \
V \ V \
(0) ----> (1) ----> (2) ----> (3) ----> (4) ----> (5)
2 \ 2 2 ^ \ 2 2 ^
\______________/ \_____________ /
3 3
factorial
method has n-1 leaves for any input n with n > 1.
Last modified 25 November 2017