Computer Science 236

Use mathematical induction to prove
(for all integers n > 0):
P(n): 1 + 3 + 6 + ... + n(n+1)/2 = n(n+1)(n+2)/6

Use mathematical induction to prove
(for all integers n > 0):
P(n): 1*1! + 2*2! + ... + n*n! = (n+1)! - 1

Consider the following facts, rules, and queries.

Facts:
  child('Jill','Zed').
  child('Ned','Bea').
  child('Tim','Jack').
  child('Sue','Jack').
  child('Anne','Jill').
  child('Lou','Jane').
  child('Mary','Tim').
  child('Ron','Dan').
  child('Anna','Kim').
  child('Tim','Jill').
  child('Mary','Jane').
  child('Jill','Bea').

Rules:
  ancestor(X,Y) :- child(Y,X).
  ancestor(X,Y) :- child(Y,Z), ancestor(X,Z).

Queries:
  ancestor('Zed','Mary')?

a. Write the rules as clauses (disjunctions of literals).

b. Write one formal proof of the query using proof-by-contradiction, instantiation, and resolution. Give a justification for each line in your proof. Each step must be a premise or the result of instantiation or resolution.

Consider the directed graph with nodes V and edges E. A diagram of the graph is provided at this link: Graph

V = { a, b, c, d, e, f }
E = { (a,b), (b,d), (c,b), (d,a), (e,d), (e,f), (f,e) }

Follow parts a-d below to show the steps of finding the strongly connected components in the graph.

a. Draw the diagram for the reverse-edge graph.

b. Draw the DFS-forest for the reverse-edge graph. (When there is a choice of nodes to visit, choose the one that is first in alphabetic order.)

c. Give the postorder numbers for the nodes in the reverse-edge graph. Assign numbers starting with the number 1.

d. Draw the DFS-tree for each strong component in the order it is found. (When there is a choice of nodes to visit, choose the one that is first in alphabetic order.)