Relations (sections 7.1

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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.

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• Equivalence relations
• Definition reflexive, symmetric and transitive.
• Equivalence class aR
• For all b, if bRa, then b ? aR
• 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.

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• 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

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• 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) 2E
• For directed graphs ?v?V deg-(v) ?v?V deg(v)
  E

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• 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

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• Types of Questions
• Conceptual
• 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/grap
  h
• Proofs
• Simple theorems or propositions
• Possible proof methods
• Induction / Direct proof /Proof by contradiction