Relations & Their Properties

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Introduction

• Let A & B be sets. • A binary relation from A to B is a subset of A x B.• Let R be a relation. If ( a, b ) R, we write a R b.• Example:

– Let S be a set of students.– Let C be a set of courses.– Let R = { (s, c) | student s is taking course c}.

• Many students may take the same course.• A single student may take many courses.

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Functions as Relations

Functions are a kind of relation.

– Let function f : A B.

– If f( a ) = b, we could write ( a, b ) f A x B.

– P( A x B ) = the set of all relations from A to B.

– Let F = the set of all functions from A to B.

– F is a proper subset of P( A x B ).

F

P( A x B )

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Relations on a Set

• A relation on a set A is a relation from A to A.

• Examples of relations on R:

– R1 = { (a, b) | a b }.

– R2 = { (a, b) | b = +sqrt( a ) }.

– Are R1 & R2 functions?

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Properties of Relations

A relation R on A is: • Reflexive: a ( aRa ).

Are either R1 or R2 reflexive?

• Symmetric: a b ( aRb bRa ).– Let S be a set of people. – Let R & T be relations on S,

R = { (a, b) | a is a sibling of b }.

T = { (a, b) | a is a brother of b }.

Is R symmetric?

Is T symmetric?

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• Antisymmetric:

1. a b ( ( aRb bRa ) ( a = b ) ).

2. a b ( ( a b ) ( ( a, b ) R ( b, a ) R ) ).

Example: L = { ( a, b ) | a b }.

Can a relation be symmetric & antisymmetric?

• Transitive:

a b c ( ( aRb bRc ) aRc ).

Are any of the previous examples transitive?

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Composition

• Let R be a relation from A to B.• Let S be a relation from B to C. • The composition is

S R = { ( a, c ) | b ( aRb bSc ) }. • Let R be a relation on A.

R1 = R

Rn = Rn-1 R.• Let R = { (1, 1), (2, 1), (3, 2), (4, 3) }.

What is R2, R3?

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End 8.1

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Graph a Relation from A to B

• The word graph above is used as a verb. • Let A = { 1, 2, 3 } and B = { 2, 3, 4 }.• Let R be a relation from A to B where { (a, b) | a

divides b }.

1 2 3A

2

3

4

B