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## Relations

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**Relations**CSC-2259 Discrete Structures Konstantin Busch - LSU**Relations and Their Properties**A binary relation from set to is a subset of Cartesian product Example: A relation: Konstantin Busch - LSU**A relation on set is a subset of**Example: A relation on set : Konstantin Busch - LSU**Reflexive relation on set :**Example: Konstantin Busch - LSU**Symmetric relation :**Example: Konstantin Busch - LSU**Antisymmetric relation :**Example: Konstantin Busch - LSU**Transitive relation :**Example: Konstantin Busch - LSU**Combining Relations**Konstantin Busch - LSU**Composite relation:**Note: Example: Konstantin Busch - LSU**Power of relation:**Example: Konstantin Busch - LSU**Theorem:**A relation is transitive if an only if for all Proof: 1. If part: 2. Only if part: use induction Konstantin Busch - LSU**1. If part:**We will show that if then is transitive Assumption: Definition of power: Definition of composition: Therefore, is transitive Konstantin Busch - LSU**2. Only if part:**We will show that if is transitive then for all Proof by induction on Inductive basis: It trivially holds Konstantin Busch - LSU**Inductive hypothesis:**Assume that for all Konstantin Busch - LSU**Inductive step:**We will prove Take arbitrary We will show Konstantin Busch - LSU**definition of power**definition of composition inductive hypothesis is transitive End of Proof Konstantin Busch - LSU**n-ary relations**An n-ary relation on sets is a subset of Cartesian product Example: A relation on All triples of numbers with Konstantin Busch - LSU**Relational data model**n-ary relation is represented with table fields R: Teaching assignments records primary key (all entries are different) Konstantin Busch - LSU**Selection operator:**keeps all records that satisfy condition Example: Result of selection operator Konstantin Busch - LSU**Projection operator:**Keeps only the fields of Example: Konstantin Busch - LSU**Join operator:**Concatenates the records of and where the last fields of are the same with the first fields of Konstantin Busch - LSU**S: Class schedule**Konstantin Busch - LSU**J2(R,S)**Konstantin Busch - LSU**Representing Relations with Matrices**Relation Matrix Konstantin Busch - LSU**Reflexive relation on set :**Diagonal elements must be 1 Example: Konstantin Busch - LSU**Symmetric relation :**Matrix is equal to its transpose: Example: For all Konstantin Busch - LSU**Antisymmetric relation :**Example: For all Konstantin Busch - LSU**Union :**Intersection : Konstantin Busch - LSU**Composition :**Boolean matrix product Konstantin Busch - LSU**Power :**Boolean matrix product Konstantin Busch - LSU**Digraphs (Directed Graphs)**Konstantin Busch - LSU**Theorem:**if and only if there is a path of length from to in Konstantin Busch - LSU**Connectivity relation:**if and only if there is some path (of any length) from to in Konstantin Busch - LSU**Theorem:**Proof: if then for some Repeated node Konstantin Busch - LSU**Closures and Relations**Reflexive closure of : Smallest size relation that contains and is reflexive Easy to find Konstantin Busch - LSU**Symmetric closure of :**Smallest size relation that contains and is symmetric Easy to find Konstantin Busch - LSU**Transitive closure of :**Smallest size relation that contains and is transitive More difficult to find Konstantin Busch - LSU**Theorem:**is the transitive Closure of is transitive Proof: Part 1: Part 2: If and is transitive Then Konstantin Busch - LSU