CSE 5211
Fall 2008
Exam 1
Points: 40
Time: 85 min
1a.
Prove by induction that the number of arcs on a tree with
n
nodes is
n
-1. [2]
1b.
Prove by induction that
[2]
1+2+3+ … n = n(n+1)/2
1c.
What is the cardinality of the set that you can form out of the subsets of a set with
n
elements? E.g., if the set is {a, b}, its subsets are {}, {a}, {b}, {a,b}, and the asked set is
{
{{}}, {{a}}, {{b}}, {{a,b}}, {{a}, {b}}, {{a}, {a,b}}, {{b}, {a,b}}, {{a}, {b}, {a,b}}
}
.
[1]
1d.
With 10 points in space (not any three of them are collinear), how many triangles can
you draw?
[1]
1e.
Find a satisfying variable assignment (a=?, b=?, c=?) for the following CNF formula:
(-a V b) ^ (b V c) ^ (-b),
Where
a, b
and
c
are Boolean variables,
–a
indicates negation of
a
, V indicates ‘or’, and ^
indicates ‘and’.
[1]
1f.
Depth of a single-node tree is 0, and the depth of any node is the depth of its parent
plus 1. Height of a tree is the maximum depth over all the nodes in the tree. Branching
factor of a balanced tree is the number of children of any node, except that of the leaves.
Draw a tree with the branching factor 3 and depth 3. What is the total number of nodes of