Discrrete mathematics for computer science digraphs and relations

Digraphs and Relations 3/22/19 Walks in digraph G walk from u to v and from v to w u v w implies walk from u to w 3/22/19 Walks in digraph G walk from u to v and from v to w, implies walk from u to w: + + u G v AND v G w + IMPLIES u G w 3/22/19 Walks in digraph G transitive relation R: u R v AND v R w IMPLIES u R w G is transitive + 3/22/19 transitivity Theorem: R is a transitive if R = G+ for some digraph G 3/22/19 Transitive Closure G+ is the transitive closure of the binary relation G 3/22/19 reflexivity relation R on set A is reflexive if a R a for all a A ≤ on numbers and ⊆ on sets are reflexive 3/22/19 reflexivity For any digraph G, * G is reflexive 3/22/19 Reflexive Transitive Closure G* is the reflexive transitive closure of the binary relation G 3/22/19 two-way walks If there is a walk from u to v and a walk back from v to u then u and v are strongly connected uG v * 3/22/19 AND vG u * 10 symmetry relation R on set A is symmetric if a R b IMPLIES b R a 3/22/19 11 Paths in DAG D path from u to v implies no path from v to u unless u=v * u D v and u≠v IMPLIES NOT(v D 3/22/19 * u) 12 antisymmetry antisymmetric relation R: u R v IMPLIES NOT(v R u) for any u ≠ v If D is a DAG then * 3/22/19 13 (weak) partial orders Reflexive, Transitive, and Antisymmetric examples: •⊆ is (weak) p.o on sets •  is (weak) p.o on  3/22/19 14 weak partial orders Theorem: R is a WPO if R = D* for some DAG D 3/22/19 15 equivalence relations transitive, symmetric & reflexive 3/22/19 16 equivalence relations Theorem: R is an equiv rel if R = the strongly connected relation of some digraph 3/22/19 17 equivalence relation examples: • = (equality) • same size • sibling (same parents) 3/22/19 18 Equivalence Relation An equivalence relation decomposes the domain into subsets called equivalence classes where aRb if a and b are in the same equivalence class In the digraph of an equivalence relation, all the members of an equivalence class are reachable from each other but not from any other equivalence class 3/22/19 19 Graphical Properties of Relations Reflexive Symmetric Transitive Equivalence Relation 3/22/19 20 Finis 3/22/19 21 ... digraph G walk from u to v and from v to w u v w implies walk from u to w 3/22/19 Walks in digraph G walk from u to v and from v to w, implies walk from u to w: + + u G v AND v G w + IMPLIES u G... 3/22/19 reflexivity relation R on set A is reflexive if a R a for all a A ≤ on numbers and ⊆ on sets are reflexive 3/22/19 reflexivity For any digraph G, * G is reflexive 3/22/19 Reflexive Transitive... relation G 3/22/19 two-way walks If there is a walk from u to v and a walk back from v to u then u and v are strongly connected uG v * 3/22/19 AND vG u * 10 symmetry relation R on set A is symmetric