summary

– Valid argument (hypotheses and conclusion) – Construction of valid argument using rules of inference
Write down each rule used, together with the statements used by the rule
Euler's formula: r= e – v + 2
– Corollary 1 – Corollary 2
Kuratowski's theorem
– Necessary and sufficient conditions for determine whether a graph is planar.
– 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
Mathematical reasoning (proofs) (section 1.5) – Rules of inference
MP, MT, resolution, simplification, addition, etc. Universal/ existential instantiation/generalization
(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.
– Partial Orderings Definition: reflexive, antisymmetric and transitive. Comparable, incomparable Totally ordered, well-ordered, lexicographic ordered Hasse Diagrams
– Closure of relations Definition: R′ is obtained by adding the minimum number of ordered pairs to R to obtain property P
– R′ R and R′ have property P – If R" R and R" have property P , then R" R′ Hence, R′ is called the P closure of R.
– 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
– Maximal and minimal elements – Greatest and least elements – Upper bound and lower bound, lub,glb
Graphs (sections 8.1 – 8.8)
– 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|
Reflexive closure r(R)= R∪△ Symmetric closure s(R)=R∪R-1 Transitive closure t(R)= R = ∪ R
∝ * n n =1
– 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:
All of its variables are instantiated All of its variables are quantified
– Nested quantifiers
Quexpressions formed by predicates, operators, and quantifiers
Functions (section 1.8) – Basics
What is a function (what are not function) Domain, co-domain, range, image, pre-image
– Types of functions
Injective (one-to-one), surjective (onto), bijective (one-to-one correspondence) Inverse function Composition of function
– Set operations (union, intersection, difference, complement)
Definitions (in words and in logical expressions) Set identity laws
– Show two sets are equal (by identity laws and by membership table)
Set Theory (sections 1.6, 1.7) – Basics
Membership, subsets, cardinality, set equality Defining sets: enumeration, builder function Cartesian product Power set
– Proof methods (proof if P then Q)
Direct proof: show if P true then Q must be true (i.e., P → Q ≡ T) Indirect proof: show that if Q is false then P must be false (its contrapositive is a tautology) Prove by contradiction: assume Q is false then derive a contradiction (i.e., derive both r and r for some r)
– Operators (, ∨, ∧, →, )
Define by truth tables Composite propositions Tautology and contradiction
– Equivalence of propositional statements
Definition Equivalence laws Proving equivalence (by truth table or equivalence laws)
– Euler and Hamilton path Definitions
– Euler path, Euler circuit, Hamilton path, Hamilton circuit
Necessary and sufficient conditions for Euler circuits and paths Sufficient conditions for Hamilton circuits – Shortest path problems Weighted graph A shortest path algorithm
– Dijkstra's algorithms
The traveling salesman problem
– approximation algorithm
– Planar graphs Definitions
– Planar graph, a planar representation of the graph
Relations (sections 7.1 – 7.5)