Discrete Structures: Understanding Relations, Study notes of Discrete Structures and Graph Theory

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CMSC 250

Discrete Structures

Relations

19 July 2007

Relations

Relations

The most basic relation is “=” (e.g. x = y)

Generally x R y

TRUE or FALSE

• R(x,y) is a more generic representation– R is a binary relation between elements of

some set A to some set B, where x

∈

A and

y∈

B

19 July 2007

Relations

Formal Definition

(Binary) relation from A to Bwhere x

A, y

B, (x,y)

A

×

B and R

A

×

B

xRy

(x,y)

R

Finite example: A={1,2}, B={1,2,3}

Infinite example: A = Z and B = Z

aRb

a-b

Z

even

19 July 2007

Relations

Example

Let A = {2,3}, B = {1,3,6}

Define a relation R from A to B such that:

xRy

↔

x – y is odd

How could this explicitly be represented as tuples?

• R = {(2,2),(2,3),(3,6)}
  

What if A and B were the set of all integers?

• R = {(x,y)

∈

Z

×Z

|

∃

k∈

Z

such that x – y = 2k + 1}

19 July 2007

Relations

Example

Define a relation of A called R

• A = {2,3,4,5,6,7,8}– R = {(4,4),(4,7),(7,4),(7,7),(2,2),(3,3),(3,6),(3,9),

(6,6),(6,3),(6,9),(9,9),(9,3),(9,6)}

Draw the arrow diagram

Is R

^

Reflexive? ^

Symmetric? ^

Transitive?

Could this arrow diagram represent a function?

19 July 2007

Relations

Example

A = {0,1,2,3}

R over A = {(0,0),(0,1),(0,3),(1,0),(1,1),(2,3),(3,0),(3,3)}

Draw the arrow diagram for it

Is R

• Reflexive?– Symmetric?– Transitive?

19 July 2007

Relations

Proving Properties on Infinite Sets

Define R(x,y) R:

Z

Z

to be

{(x,y)

Z

×

Z

| x|y}

Prove whether or not this is:

• Reflexive?– Symmetric?– Transitive?

19 July 2007

Relations

The Inverse of the Relations

and the Complement of the Relation

R:

A

→

B

R

= {(x,y)

∈

A

×

B | xRy}

R

:B

→

A

R

= {(y,x)

∈

B

×

A | (x,y)

∈

R}

R

’: A

→

B

R

’^

= {(x,y)

∈

A

×

B | (x,y)

∉

R}

D: {1,2}

→

{2,3,4}

D={(1,2),(2,3),(2,4)}

D

’^

=?

D

=?

S = {(x,y)

∈

R

×

R | y = 2*|x|}

S

=?

R y x R y x Y y X x

,^

'

R y x R x y Y y X x

−

,^

1

19 July 2007

Relations

Example

Given the relation R = {(0,1),(1,2),(2,3)}

What is R

RC

What is R

SC

What is R

TC

19 July 2007

Relations

Matrix Representation

of a Relation

M

R^

= [m

]ij

-^

m

={1 iff (i,j)ij

∈

R and 0 iff (i,j)

∉

R}

Example:

-^

R : {1,2,3}

→

{1,2}

R = {(2,1),(3,1),(3,2)}

   

   

=

0 0 1 0 1 1 1 2 3

2 1

R M

19 July 2007

Relations

Example

Let ID = set of student IDs

Let Class = set of classes offered

Define relation Summer2007^ {(x,y)

∈

ID

×Class | student x is registered for class y}

Can we do this?

Relational databases …

19 July 2007

Relations

Binary Relation Induced by a Partition

Let A1,A2,…,An be a partition of A

Let relation R: A

×

A be defined as:

{(x,y)

∈

A

×

A |

∃

i 1

≤

i≤

n x

∈

A

∧i

y

∈

A

}i

A = {1,2,3,4,5,6,7}

A

1

= {1,2,3}; A

2

=

{4,5}

; A

3

=

{6,7}

R = {(1,1),(2,2),(3,3),(1,2),(1,3),(2,3),(2,1),(3,2),(3,1),^ (4,4),(4,5),(5,4),(5,5)

,(6,6),(6,7),(7,6),(7,7)

}

Note that this (or any partition-induced relation) isreflexive, symmetric, and transitive

19 July 2007

Relations

Examples

R: X

X

X={a,b,c,d,e,f}

{(a,a),(b,b),(c,c),(d,d),(e,e),(f,f),(a,e),(a,d),(d,a)(d,e),(e,a),(e,d),(b,f),(f,b)}

Lemma 10.3.

If A is a set and R is an

equivalence relation on A and x and y areelements of A, then either

[x]

[y] =

or

[x]=[y]

19 July 2007

Relations

Example

R(x,y); R:

Z

Z

{(x,y)

Z

×

Z

| 3|(x – y)}

How many equivalence classes are there?

[a] = {x

Z

| R(x,a)}

= {x

Z

n

Z

such that x = 3n + a}

Classes are: [0], [1], [2]