Discrete Mathematics Final Review
I.
All Previous Review Sheets
I.
Graph
A.
Definition of G = (V, E)
1.
Digraph
a.
Directed edges
2.
Bipartite
3.
Edges
a.
Directed
b.
Paths
i.
Euler path
ii. Simple pa
Discrete Mathematics Review 2
Recursively Defined Objects
A.
Recursive Definition
1.
Basic Step
2.
Recursive Step
B.
Examples
1.
Binary Trees
2.
Rooted Trees
3.
Concatenated Strings
C.
Structural Indu
Discrete Mathematics Review 1
I.
Logic
A.
Proposition
B.
Logical Operators
1.
And
2.
Or
3.
Xor
4.
Not
5.
Implies
a.
b.
Hypothesis and Conclusion
c.
Truth Table
d.
6.
In terms of Or and Not
Converse, I
Discrete Mathematics Review 2
0.
I.
Recurrence Relations (from last review sheet)
A.
Constructing sequences
1.
Towers of Hanoi
2.
Strings
3.
Fibonacci sequence
B.
Difference equations as recurrence re
Discrete Mathematics Review 2
I.
Mathematical Induction
A.
Steps in mathematical induction
1.
P(1)
2.
P(n) P(n + 1)
B.
Strong Induction
1.
uses all the P(m) for m < n
C.
Well-Ordering Principle
1.
Min
Examination 2 Solutions Discrete Mathematics
1.
Are the following pair of graphs isomorphic? WHY?
u1
u2
u3
u5
u6
u4
u8
u7
v1
v2
v4
v3
v5
v6
v8
v7
SOLUTION. The graphs are not isomorphic. One way of se
Solutions Discrete Mathematics Examination 2
1.
Determine whether the relation with the following graph is an equivalence relation.
a
b
c
d
SOLUTION. The graph does not represent an equivalence relati
Solutions Examination 2 Matrix Methods
1.
Find the characteristic polynomial and the eigenvalues of the matrix A =
2 0 -1
121.
-1 0 2
SOLUTION. The characteristic polynomial is
cA HxL = Det
2-x
0
-1
2
Review Sheet 2 Matrix Methods
I.
II.
Determinants
A. Three Defining Properties
1. Det I = 1
2. Multilinear
3. Anti symmetric (interchanging two rows produces a minus sign)
B. Evaluation by Cofactors
1
Solutions Examination 2 Matrix Methods
1a.
Find the determinant of the matrix A =
1 1 2 -2
1 5 2 -1
.
-2 -2 1 3
-3 4 -1 8
SOLUTION. The determinant is 25. The problem can be done by hand as
1
1
-2
-3