Module #21 - Relations

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8.3 Representing Relations

Rosen 6Rosen 6thth ed., Ch. 8 ed., Ch. 8

Module #21 - Relations

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§8.3: Representing Relations

• Some ways to represent Some ways to represent nn-ary relations:-ary relations:– With an explicit list or table of its tuples.With an explicit list or table of its tuples.– With a function from the domain to With a function from the domain to {{TT,,FF}}..

• Or with an algorithm for computing this function.Or with an algorithm for computing this function.

• Some special ways to represent binary Some special ways to represent binary relations:relations:– With a zero-one matrix.With a zero-one matrix.– With a directed graph.With a directed graph.

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Using Zero-One Matrices• To represent a binary relation To represent a binary relation RR::AA××BB by an by an ||AA||×|×|BB||

0-1 matrix 0-1 matrix MMRR = [ = [mmijij]], let , let mmijij = 1 = 1 iff iff ((aaii,,bbjj))RR..• E.g.E.g., Suppose Joe likes Susan and Mary, Fred , Suppose Joe likes Susan and Mary, Fred

likes Mary, and Mark likes Sally.likes Mary, and Mark likes Sally.• Then the 0-1 matrix Then the 0-1 matrix

representationrepresentationof the relationof the relationLikes:BoysLikes:Boys×Girls×Girlsrelation is:relation is:

1 00

010

0 1 1

Mark

Fred

JoeSallyMarySusan

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Properties of Relations

• Reflexivity: A relation R on A Reflexivity: A relation R on A x x A is A is reflexive if for all reflexive if for all aaA, (A, (a,aa,a) R.) R.

• Symmetry: A relation R on AxA is Symmetry: A relation R on AxA is symmetric if (symmetric if (x,yx,y) R implies () R implies (y,xy,x) R. ) R.

• Anti-symmetry:Anti-symmetry: A relation on A A relation on A xx A is anti-symmetric if A is anti-symmetric if

((a,ba,b) R implies () R implies (b,ab,a) R. Or a = b.) R. Or a = b.

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Zero-One Reflexive, Symmetric

• Terms:Terms: Reflexive Reflexive, s, symmetric, and ymmetric, and antisymmetricantisymmetric..– These relation characteristics are very easy to These relation characteristics are very easy to

recognize by inspection of the zero-one matrix.recognize by inspection of the zero-one matrix.

€

1

1

1

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

1

1 0

0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

0

1 0 1

0

0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Reflexive:all 1’s on diagonal

Symmetric:all identical

across diagonal

Antisymmetric:all 1’s are across

from 0’s

any-thing

any-thing

anything

anything

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Using Directed Graphs

• A A directed graphdirected graph or or digraphdigraph GG=(=(VVGG,,EEGG)) is a set is a set VVGG of of

vertices (nodes)vertices (nodes) with a set with a set EEGGVVGG××VVGG of of edges edges

(arcs,links)(arcs,links). Visually represented using dots for nodes, . Visually represented using dots for nodes, and arrows for edges. Notice that a relation and arrows for edges. Notice that a relation RR::AA××BB can be can be represented as a graph represented as a graph GGRR=(=(VVGG==AABB, , EEGG==RR))..

1 00

010

0 1 1

Mark

Fred

JoeSallyMarySusan

Matrix representation MR: Graph rep. GR: Joe

Fred

Mark

Susan

Mary

Sally

Node set VG

(black dots)

Edge set EG

(blue arrows)

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Digraph Reflexive, Symmetric

It is extremely easy to recognize the reflexive/irreflexive/ It is extremely easy to recognize the reflexive/irreflexive/ symmetric/antisymmetric properties by graph inspection.symmetric/antisymmetric properties by graph inspection.

Reflexive:Every node

has a self-loop

Irreflexive:No node

links to itself

Symmetric:Every link isbidirectional

Antisymmetric:

No link isbidirectional

These are asymmetric & non-antisymmetric These are non-reflexive & non-irreflexive

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Properties of Relations

• Transitivity:Transitivity:

A relation on A A relation on A xx A is transitive if ( A is transitive if (a,ba,b) ) R R and (and (b,cb,c) ) R imply ( R imply (a,ca,c) ) R. R.

Graph??Graph??