CS/EE 5740/6740:Computer Aided Design of Digital Circuits

Chris J. Myers

Lecture 3: Sets, Relations, andFunctions

Reading: Chapter 3.1

∅

Set Notation

| |V Cardinality (number of members) of set V

v V∈ Element is a member of set v V

v V∉ Element is not a member of set v V

S V⊆ Set is a subset of set S V

The empty set (a member of all sets)

′S The complement of set S

VS

U

U The universe: ′ = −S U S

Power Sets

2V The power set of set (the set of allsubsets of set

VV

| | | |2 2V V= The cardinality of a power set is apower of 2

2V S S V= ⊆{ | }

Power Sets

V = { , , }012

2

0 1 2

01 0 2 12

012

V = ∅{

}

,

{ } ,{ } ,{ } ,

{ , } ,{ , } ,{ , } ,

{ , , }

3-member set

1 subset with 0 members

3 subsets with 1 members

3 subsets with 2 members

1 subset with 3 members

| | | |2 2 2 83�

V V= = =

Power sets are Boolean Algebras

Cartesian Products

• The Cartesian Product of sets and isdenoted

• Suppose , then

A B×BA

A B a b= ={ , , } , { , }012

A B a b a b a b× = { ( , ),( , ),( , ),( , ),( , ),( , )}0 0 1 1 2 2

A = { , , }012 Set is unordered( ) denotes Ordered Set

A( , )1 b

Binary Relations

• The Cartesian Product of sets and isdenoted

• consists of all possible orderedpairs such that and

• A subset is called a BinaryRelation

• Graphs, Matrices, and Boolean Algebrascan be viewed as binary relations

A B×BA

A B×( , )a b a A∈ b B∈

R A B⊆ ×

Binary Relations as Graphs or Matrices

1 0�

0�

1 1 1

Rectangular Matrix

A a c= { , } B b d f= { , , }

E ab cb cd cf A B= ⊆ ×{ , , , }

a b

c d

fa

b

c

d f

Bipartite Graph

1 1 1 1

0 1 0 1

0 0 1 1

0 0 0 1

Square Matrix

a

cb

d

V a b c d= { , , , }

E ab ac ad bd cd aa bb cc dd V V= ∪ ⊆ ×{ , , , , } { , , , }

a

c

b

d

Directed Graph

Binary Relations as Graphs or Matrices

If , we say that is the domainof the relation , and that is the range.

Notation for Binary Relations

R A B⊆ × ABR

If , we say that the pairis in the relation , or .

( , )a b R∈R aRb

Example: “Less than or Equal”

A B

R A B

= =⊆ × = ≤

{ , , , } ,

" "

012 �

≤ ≤ ≤ ≤≤ ≤ ≤

≤ ≤≤

0

0 0

0 0 0

0

1

2

3

0 1 2 3

Example: “a times b = 120”

V

R V V u v u v

R

=⊆ × = × ==

{ , , , } ,

{ ( , )| }

{ ( , ),( , ),( , ),( , ),( , ),( , )}

012

12

112 2 6 3 4 4 3 6 2 121

1

12

2 3

46

Properties of Binary Relations

• A binary relation can be– reflexive, and/or

– transitive, and/or

– symmetric, and/or

– antisymmetric

• We illustrate these properties on the nextfew slides

R V V⊆ ×

Reflexive BinaryRelations

A binary relation is reflexive if and only if for every vertex

1 1 1 1

0 1 0 1

0 0 1 1

0 0 0 1

a

c

b

d

a

cb

d

V a b c d= { , , , }

R V V

v V vRv

⊆ ×∈ ⇒

R V V⊆ ×( , )v v R∈

v V∈

Non-ReflexiveBinary Relations

1 1 1 1

0 1 0 1

0 0 1 1

0 0 0 0

a

c

b

d

a

cb

d

V a b c d= { , , , }

R V V

v V vRv

⊆ ×∃ ∈ ∋ ¬

Non-Reflexiviety implies thatthere exists such that .

v V∈( , )v v R∉

Transitive Binary Relations

a

b� c�

d�

A binary relation is transitive if and onlyif every path is triangulated by adirect edge.

( , )u v

( , )u v

e

If and

and then

u v w V uRv

vRw uRw

, , , ,

, .

∈

Non-Transitive Binary Relations

A binary relation is not transitive if thereexists a path from to that is nottriangulated by a direct edge.

R V V

u v w V

uRv vRw vRv

⊆ ×∃ ∈

¬, , ,

, ,

such that

but

a

b� c�

d�

e

( , )u vu v

A binary relation is symmetric if and onlyif every edge isreciprocated by a edge

Symmetric Binary Relations

( , )u v

( , )v u

R V V

u v R v u R

uRv vRu

⊆ ×∈ ⇒ ∈

⇒( , ) ( , )

( )

1

12

2 3

46

A binary relation is non-symmetric if thereexists an edge notreciprocated by anedge

Non-Symmetric Binary Relations

( , )u v

( , )v u

R V V

u v R v u R

⊆ ×∃ ∈ ∉( , ) ( , )such that

ed

cb

a

Antisymmetric Binary Relations

∀ ∈ ⇒ =( , )�

, ( , )�

( )�

u v R

u R

v v R

u v u

A binary relation is anti-symmetric if and onlyif no edge isreciprocated by aedge

( , )u v( , )v u

Not antisymmetric if any such edge is reciprocated.

v u=

ed

cb

a

Equivalence Relations

Note R is reflexive, symmetric, and transitive

a

b�

c�

d�

eed

cb

a

The Path Relation

This graph defines pathrelation * →

ed

cb

a

G V E=( , )

* →

a a b c d eb c d ec c ed c ee c e

*

*

*

*

*

{ , , , , }{ , , }{ , }{ , }{ , }

→

→

→

→

→

* →Relation is sometimes called “Reachabilit y”

Equivalence Relations

This graph defines pathrelation This graph defines R

NOT an equivalencerelation

An equivalencerelation

* →

ed

cb

a

ed

cb

a

R u v v u u vG V R

= → →

={ ( , )| , }* *

( , )G V E=( , )

The Cycle Relation

This graph defines pathrelation *← →

ed

cb

a

ed

cb

a

C u v v uG V R

= ← →

={ ( , )| }*

( , )G V E=( , )

*← →

These are called the “StronglyConnected Components” of G

Functions

• A function f is a binary relation from set A(called the domain) to B (called the range)

• But, it is required that each a in A beassociated with exactly 1 b in B

• For functions, it cannot be true that both(a,b) in R and (a,c) in R, b different from c

Image and Preimage

• Consider a function f(x) : A -> B:

• Given a doman C A, the set

IMG(f,C) = { y B | x C s.t. y=f(x)}

is called the image of C under the mapping f.

• Given a range C B, the set

PRE(f,C) = { x A | y C s.t. y=f(x)}

is called the preimage of C under f.

⊆

⊆

∃

∃

∈∈

∈ ∈

Properties of Functions

• A function f is one-to-one (or injective), ifx=y implies f(x)=f(y).

• A function f is onto (or surjective) if, forevery y B, there exists an element x A,such that f(x)=y.

• A function that is both one-to-one and ontois called bijective, and is invertible.

∈∈

Other Binary Relations

• Compatibilit y Relations (Ch.: 8 FSM Minimization)•Reflexive•Symmetric•Not transitive

•Partial Orders (Includes Lattices, Boolean Algebras)•Reflexive•Transitive•Antisymmetric

The Binary Relation of Relationsto Synthesis/Verification

Equivalence Relations

Partial Orders

Compatibili ty Relations

Sequential Logic (no DCs)

Sequential Logic (with DCs)

*DC�

=don�

’ t c� a� re�

Combinational Logic (no DCs)=> (0,1) Boolean Algebra

Combinational Logic (with DCs)=> Big Boolean Algebras