CSC-2259 Discrete Structures 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

A relation is transitive if an only if for all Theorem: A relation is transitive if an only if for all Proof: 1. If part: 2. Only if part: use induction Konstantin Busch - LSU

Definition of composition: 1. If part: We will show that if then is transitive Assumption: Definition of power: Definition of composition: Therefore, is transitive Konstantin Busch - LSU

We will show that if is transitive then for all 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

End of Proof 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

(all entries are different) Relational data model n-ary relation is represented with table fields R: Teaching assignments Professor Department Course-number Cruz Zoology 335 412 Farber Psychology 501 617 Rosen Comp. Science 518 Mathematics 575 records primary key (all entries are different) Konstantin Busch - LSU

keeps all records that satisfy condition Selection operator: keeps all records that satisfy condition Example: Result of selection operator Professor Department Course-number Farber Psychology 501 617 Konstantin Busch - LSU

Keeps only the fields of Projection operator: Keeps only the fields of Example: Professor Department Cruz Zoology Farber Psychology Rosen Comp. Science Mathematics Konstantin Busch - LSU

Concatenates the records of and where the last fields of 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 Department Course-number Room Time Comp. Science 518 2:00pm Mathematics 575 N502 3:00pm 611 4:00pm Psychology 501 A100 617 A110 11:00am Zoology 335 9:00am 412 8:00am Konstantin Busch - LSU

J2(R,S) Professor Department Course Number Room Time Cruz Zoology 335 9:00am 412 8:00am Farber Psychology 501 3:00pm 617 A110 11:00am Rosen Comp. Science 518 N521 2:00pm Mathematics 575 N502 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

Matrix is equal to its transpose: 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

Boolean matrix product Composition : Boolean matrix product Konstantin Busch - LSU

Boolean matrix product Power : Boolean matrix product Konstantin Busch - LSU

Digraphs (Directed Graphs) Konstantin Busch - LSU

there is a path of length from to in 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

Smallest size relation that contains and is symmetric 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

is the transitive Closure of Theorem: is the transitive Closure of is transitive Proof: Part 1: Part 2: If and is transitive Then Konstantin Busch - LSU