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.

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.

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

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|

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

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