Partial Orders Strict Orders Equivalence Relations Sections 6.7-6.9 Prof. Sandy Irani Partial Orders A binary relation R on a set A is a partial order if R is: • Reflexive • Anti-Symmetric • Transitive Notation: x, y ∈ A, xRy ↔ x ≼ y Standard example: (ℤ, ≤) x is related to y if x ≤ y Reflexive? Anti-symmetric? Transitive? Partial Orders + Another example: (ℤ , ≼푝) + + n For x, y ∈ ℤ , x ≼푝 y if there is an n ∈ ℤ such that x = y Relfexive? Anti-Symmetric? Transitive? Incomparable Elements Another example: ({2, 4, 8, 64}, ≼푝) + + n For x, y ∈ ℤ , x ≼푝 y if there is an n ∈ ℤ such that x = y 4 4 ≼푝8? 2 64 8 ≼푝 4? 8 4 and 8 are incomparable Two elements, x and y, in a partial order are incomparable if x ⋠ y and y ⋠ x Incomparable Elements What about (ℤ, ≤)? Are any two elements incomparable? 4 ≤ 8? 8 ≤ 4? 4 ≤ 4? A partial order (A, ≼) is also a total order if for every x, y ∈ A, x ≼ y or y ≼ x. + (ℤ , ≼푝) is not a total order Partial Orders/Total Orders Relation R on the set {0,1}5 is defined as follows. For every x, y ∈ {0,1}5 x is related to y if y can be obtained by taking x and changing zero or more 0’s to 1’s. Examples: Partial Orders/Total Orders Relation R on the set {0,1}5 is defined as follows. For every x, y ∈ {0,1}5 x is related to y if y can be obtained by taking x and changing zero or more 0’s to 1’s. Reflexive? Anti-symmetric? Transitive? Total Order? Prefix A string x is a prefix of a string y if y can be obtained by adding zero or more characters to the end of x. Consider {a, b}* (the set of all strings over the alphabet {a, b}.) Examples: Partial Orders/Total Orders * Relation R on the set {a, b}* is defined as follows. For every x, y ∈ {a, b}* xRy if x is a prefix of y Select the correct description of R. A) Partial Order C) Total Order and a Total Order but not a Partial Order B) Partial Order D) Neither a Total Order but not a Total Order nor a Partial Order Partial Orders/Total Orders Relation R on the set {a, b}* is defined as follows. For every x, y ∈ {a, b}* xRy if x is a prefix of y Reflexive? Anti-symmetric? Transitive? Total Order? Partial Orders/Total Orders * Relation R on the set of players on a football team. For every two players, x and y on the team, xRy if y weighs at least as much as x (Big team, so can assume that here are all possible height/weight combinations) Select the correct description of R. A) Partial Order C) Total Order and a Total Order but not a Partial Order B) Partial Order D) Neither a Total Order but not a Total Order nor a Partial Order Partial Orders/Total Orders Relation R on the set of players on a football team. For every two players, x and y on the team, xRy if y weighs at least as much as x Reflexive? Anti-symmetric? Transitive? Total Order? Partial Orders/Total Orders Relation R on the set of players on a football team. For every two players, x and y on the team, xRy if y weighs at least as much as x (No two players on the team have the same weight) Reflexive? Anti-symmetric? Transitive? Total Order? Hasse Diagrams Hasse Diagrams are a clean way to depict a partial order. If x ≼ y, then x appears below y. (Converse not necessarily true) A line between x and y if x ≼ y and there is no z such that x ≼ z and z ≼ y Hasse Diagrams If x ≼ y, then x appears ( {2, 4, 8, 16, 32, 64}, ≼푝) below y. (Converse not necessarily true) A line between x and y if x ≼ y and there is no z such that x ≼ z and z ≼ y Maximal/Minimal Elements in a Partial Order x is a maximal element ( {2, 4, 8, 16, 32, 64}, ≼푝) if there is no y such that 64 x ≼ y 16 x is a minimal element 8 32 4 if there is no y such that y ≼ x 2 Maximal elements: Minimal elements: Hasse Diagrams * g e Which pair of elements are not comparable? a c A) b and e b B) f and e d f C) b and c D) b and g Hasse Diagrams The Hasse diagram for a total order is just a chain: ( {1, 2, 3, 4, 5}, ≤ ) 5 4 3 2 1 Strict Orders A binary relation R on a set A is a strict order if R is: • Anti-Reflexive • Anti-Symmetric • Transitive Notation: x, y ∈ A, xRy ↔ x ≼ y Standard example: (ℤ, <) x is related to y if x < y Anti-Reflexive? Anti-symmetric? Transitive? Strict vs. Partial Orders Partial Order Strict Order 4 4 2 64 2 64 8 8 For x, y ∈ ℤ, x ≼푝 y For x, y ∈ ℤ, x ≼푝 y if there is an n, if there is an n, n ∈ ℤ+ n ∈ ℤ+, n > 1 and xn = y and xn = y Strict Orders A binary relation R on a set A is a strict order if R is: • Anti-Reflexive • Transitive If a relation R is anti-reflexive and transitive, then R is also anti-symmetric. [ (Anti-Reflexive) ᴧ (Transitive) ] → (Anti-Symmetric) Contrapositive: [¬(Anti-Symmetric) ᴧ (Transitive) ] → ¬(Anti-Reflexive) [¬(Anti-Symmetric) ᴧ (Transitive) ] → ¬(Anti-Reflexive) Proof: Anti-symmetric means you never have this pattern: ¬(Anti-Symmetric) means that you do have this pattern Somewhere: If the relation is transitive, then you also have: Which means that the relation is not Anti-Reflexive. □ Strict Orders A binary relation R on a set A is a strict order if R is: • Anti-Reflexive • Transitive A strict order is also a total order if for every x, y if x ≠ y then xRy or yRx (i.e. every distinct pair of elements in the domain are comparable.) Partial Orders/Total Orders * Relation R on the set {0,1}8 is defined as follows. For every x, y ∈ {0,1}8 if y contains more 1’s than x. Select the correct description of R. A) Strict Order C) Total Order and a Total Order but not a Strict Order B) Strict Order D) Neither a Strict Order but not a Total Order nor a Partial Order Partial Orders/Total Orders Relation R on the set {0,1}8 is defined as follows. For every x, y ∈ {0,1}8 if y contains more 1’s than x. Anti-Reflexive? Anti-symmetric? Transitive? Total Order? Partial Orders/Total Orders Relation R on a group of people. For every x, y in the group, xRy if y is taller than x. Select the correct description of R. Anti-Reflexive? Anti-symmetric? Transitive? Total Order? Precedence Relationships Strict orders are useful for representing precedence relationships Vertices: set of tasks Relation: task x must be completed before task y begins. Prerequisite structure for some ICS/CS classes: ICS 31 ICS 32 ICS 33 ICS 51 CS 151 CS 152 ICS 6B ICS 6D Prerequisite are represented by a directed graph with no cycles which is not necessarily transitive. Directed Acyclic Graphs (DAGs) A directed graph is acyclic if it does not have any positive length cycles. <v> is a cycle of length 0. (no edges) <v,v> is a cycle of length 1. v DAGs and Precedence Constraints Graph representing prerequisite structure must be a DAG G. Course x must be taken before course y ↔ (x,y) is in the transitive closure of G. Theorem: G is a directed acyclic graph (DAG) if and only if G+ is a strict order. Topological Sort of a DAG A topological sort of a DAG G is an ordering of the vertices such that for every edge (u,v) in G, u comes before v in the ordering. ICS 31 ICS 32 ICS 33 ICS 51 CS 151 CS 152 ICS 6B ICS 6D Note: there can be more than one topological sorts for a graph. Topological Sorts * Which ordering is not a valid topological sort for the graph given below: d a f e b c A) f, d, a, c, b, e C) f, b, d, a, e, c B) f, c, d, b, a, e Topological Sorts To find a topological sort, keep removing vertices that are minimal (in-degree 0) d a f e b c Equivalence Relations A binary relation R is an equivalence relation if R is: • Reflexive • Symmetric • Transitive Notation: if xRy then x ~ y a d e c b f Equivalence Relation Example Domain: ℤ For x, y ∈ ℤ, x ~ y if x-z is an integer multiple of 5 (x - z = 5k for k ∈ ℤ) Here is the arrow diagram for the relation on a restricted, finite set: {-4, -1, 2, 6, 7, 8, 19, 21, 27} 27 19 -1 6 7 8 2 21 -4 Equivalence Relation Example Domain: ℤ For x, y ∈ ℤ, x ~ y if x-z is an integer multiple of 5 (x - z = 5k for k ∈ ℤ) Reflexive: Symmetric Transitive: Equivalence Relation Example Domain: group of people For x, y in the group, x ~ y if x and y have the same birthday Reflexive: Symmetric Transitive: Equivalence Relations * Domain is a group of people: Relation R: xRy if x and y have the same first name or same last name.