1. Relations (sections 7.1 – 7.5)
Summary 3
Relations (sections 7.1 – 7.5)
Definitions.
(Binary) relation R from A to B: R  AB: ordered pairs
Relation R on A: R  AA
n-ary relation R  A1A2…An
Functions are special cases of relations. (What are the differences?)
Properties of relations
Reflexive/irreflexive
Symmetric/asymmetric/antisymmetric
Transitive
Properties reflected in graph and matrix representations
Combining relations: R1  R2, R1  R2, R1 – R2, SR.

2. Equivalence relations Definition: reflexive, symmetric and transitive.
Equivalence class: [a]R
For all b, if bRa, then b  [a]R
aRb iff [a] = [b] iff [a]  [b]  
not aRb iff [a]  [b] iff [a]  [b] = 
Partition of a set:
(i) Ai   for iI
(ii) Ai  Aj = , if i  j
(iii) iI Ai = S
All equivalence classes of R on A partition A.

3. Representing relations: R from A to B
Set of ordered pairs:
{(a, b)| aRb for (a, b)  AB}
0 – 1 matrix
mij = 1, if (ai, bj)R, and
mij = 0, if (ai, bj)R.
Directed graph: (V, E)
V = A  B ( or V = A if R is on A)
(a, b)  E iff (a, b) R
Be able to convert between the three representations

4. Graphs (sections 8.1 – 8.4) Definitions.
Simple, undirected and directed graphs.
Degree (indegree, outdegree) of vertex
Loop, isolated and pendant vertices
Special graphs
Complete, bipartite, and complete bipartite, n-cube
Subgraph
Adjacency
For undirected graphs:vV deg(v) = 2|E|
For directed graphs: vV deg-(v) = vV deg+(v) = |E|

5. Representing graphs
Adjacency lists
Adjacency matrix
aij = 1 if {vi, vj} is an edge of G; and
aij = 0 otherwise.
Connectivity
Path, path length, simple path, circuit
A (undirected) graph is connected if there is a path between any pair of vertices
Strongly and weakly connected directed graphs
Connected components of a graph

6. Types of Questions Conceptual Problem solving Proofs
Definitions of terms
True/false
Simple questions
Problem solving
Work with small concrete example problems
Use your knowledge in a comprehensive way for problem-solving:
set/subset/Cartesian product/relation/matrix/graph
Proofs
Simple theorems or propositions
Possible proof methods
Induction / Direct proof /Proof by contradiction