### Discrete Mathematics

WEN-CHING LIEN Department of Mathematics National Cheng Kung University

2008

### 7.1: Relations Revisited: Properties of Relations

Definition (7.1)

*For sets A, B, any subset of A*×*B is called a (binary) relation*
from A to B.

*Any subset of A*×*A is called a (binary) relation on A.*

WEN-CHINGLIEN **Discrete Mathematics**

### 7.1: Relations Revisited: Properties of Relations

Definition (7.1)

*For sets A, B, any subset of A*×*B is called a (binary) relation*
from A to B.

*Any subset of A*×*A is called a (binary) relation on A.*

Example (7.1)

a) Define the relation R on the set Z by aRb, or(a,*b) ∈R,*if
*a*≤*b.This subset of Z* ×*Z is the ordinary ”less than or*
equal to” relation on the set Z, and it can also be defined
on Q or R, but not C.

b) *Let n*∈*Z*^{+}.*For x,y* ∈*Z*,the modulo n relation R is
*defined by xRy if x* −*y is a multiple of n.With n*=7, we
*find, for instance, that 9R2,*−3R11,(14,0) ∈*R,*but 3|R7
(that is , 3 is not related to 7).

...continued

WEN-CHINGLIEN **Discrete Mathematics**

Example (7.1)

a) Define the relation R on the set Z by aRb, or(a,*b) ∈R,*if
*a*≤*b.This subset of Z* ×*Z is the ordinary ”less than or*
equal to” relation on the set Z, and it can also be defined
on Q or R, but not C.

b) *Let n*∈*Z*^{+}.*For x,y* ∈*Z*,the modulo n relation R is
*defined by xRy if x* −*y is a multiple of n.With n*=7, we
*find, for instance, that 9R2,*−3R11,(14,0) ∈*R,*but 3|R7
(that is , 3 is not related to 7).

...continued

Example (7.1)

a) Define the relation R on the set Z by aRb, or(a,*b) ∈R,*if
*a*≤*b.This subset of Z* ×*Z is the ordinary ”less than or*
equal to” relation on the set Z, and it can also be defined
on Q or R, but not C.

b) *Let n*∈*Z*^{+}.*For x,y* ∈*Z*,the modulo n relation R is
*defined by xRy if x* −*y is a multiple of n.With n*=7, we
*find, for instance, that 9R2,*−3R11,(14,0) ∈*R,*but 3|R7
(that is , 3 is not related to 7).

...continued

WEN-CHINGLIEN **Discrete Mathematics**

Example (7.1)

*b) ∈R,*if
*a*≤*b.This subset of Z* ×*Z is the ordinary ”less than or*
equal to” relation on the set Z, and it can also be defined
on Q or R, but not C.

*Let n*∈*Z*^{+}.*For x,y* ∈*Z*,the modulo n relation R is
*defined by xRy if x* −*y is a multiple of n.With n*=7, we
*find, for instance, that 9R2,*−3R11,(14,0) ∈*R,*but 3|R7
(that is , 3 is not related to 7).

...continued

Example (7.1 continued)

c) *For the universe U* = {1,2,3,4,5,6,7}consider the (fixed)
*set C* ⊆*U where C* = {1,2,3,6}.Define the relation R on
*P(U)by ARB when A*∩*C*=*B*∩*C.*Then the sets

{1,2,4,5}and{1,2,5,7}are related since

{1,2,4,5} ∩*C*= {1,2} = {1,2,5,7} ∩*C.*Likewise we find
*that X* = {4,5}*and Y* = {7}are so related because
*X* ∩*C* = ∅ =*Y* ∩*C.However, the sets S* = {1,2,3,4,5}

*and T* = {1,2,3,6,7}are not related. Since
*S*∩*C*= {1,2,3} 6= {1,2,3,6} =*T* ∩*C.*

WEN-CHINGLIEN **Discrete Mathematics**

Example (7.1 continued)

c) *For the universe U* = {1,2,3,4,5,6,7}consider the (fixed)
*set C* ⊆*U where C* = {1,2,3,6}.Define the relation R on
*P(U)by ARB when A*∩*C*=*B*∩*C.*Then the sets

{1,2,4,5}and{1,2,5,7}are related since

{1,2,4,5} ∩*C*= {1,2} = {1,2,5,7} ∩*C.*Likewise we find
*that X* = {4,5}*and Y* = {7}are so related because
*X* ∩*C* = ∅ =*Y* ∩*C.However, the sets S* = {1,2,3,4,5}

*and T* = {1,2,3,6,7}are not related. Since
*S*∩*C*= {1,2,3} 6= {1,2,3,6} =*T* ∩*C.*

Example (7.1 continued)

c) *For the universe U* = {1,2,3,4,5,6,7}consider the (fixed)
*set C* ⊆*U where C* = {1,2,3,6}.Define the relation R on
*P(U)by ARB when A*∩*C*=*B*∩*C.*Then the sets

{1,2,4,5}and{1,2,5,7}are related since

{1,2,4,5} ∩*C*= {1,2} = {1,2,5,7} ∩*C.*Likewise we find
*that X* = {4,5}*and Y* = {7}are so related because
*X* ∩*C* = ∅ =*Y* ∩*C.However, the sets S* = {1,2,3,4,5}

*and T* = {1,2,3,6,7}are not related. Since
*S*∩*C*= {1,2,3} 6= {1,2,3,6} =*T* ∩*C.*

WEN-CHINGLIEN **Discrete Mathematics**

Example (7.1 continued)

*For the universe U* = {1,2,3,4,5,6,7}consider the (fixed)
*set C* ⊆*U where C* = {1,2,3,6}.Define the relation R on
*P(U)by ARB when A*∩*C*=*B*∩*C.*Then the sets

{1,2,4,5}and{1,2,5,7}are related since

*C*= {1,2} = {1,2,5,7} ∩*C.*Likewise we find
*that X* = {4,5}*and Y* = {7}are so related because
*X* ∩*C* = ∅ =*Y* ∩*C.However, the sets S* = {1,2,3,4,5}

*and T* = {1,2,3,6,7}are not related. Since
*S*∩*C*= {1,2,3} 6= {1,2,3,6} =*T* ∩*C.*

Example (7.1 continued)

*For the universe U* = {1,2,3,4,5,6,7}consider the (fixed)
*set C* ⊆*U where C* = {1,2,3,6}.Define the relation R on
*P(U)by ARB when A*∩*C*=*B*∩*C.*Then the sets

{1,2,4,5}and{1,2,5,7}are related since

*C*= {1,2} = {1,2,5,7} ∩*C.*Likewise we find
*that X* = {4,5}*and Y* = {7}are so related because
*X* ∩*C* = ∅ =*Y* ∩*C.However, the sets S* = {1,2,3,4,5}

*and T* = {1,2,3,6,7}are not related. Since
*S*∩*C*= {1,2,3} 6= {1,2,3,6} =*T* ∩*C.*

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.2)

A relation R on a set A is called reflexive if for all
*x* ∈*A,*(x,*x) ∈R.*

Example (7.4)

*For A*= {1,2,3,4},*a relation R*⊆*A*×*A will be reflexive if and*
*only if R* ⊇ {(1,1),(2,2),(3,3),(4,4)}.

*Consequently, R*_{1}= {(1,1),(2,2),(3,3)}is not a reflexive
*relation on A, whereas R*_{2}= {(x,*y*)|x,*y* ∈*A,x* ≤*y}*is reflexive
on A.

Definition (7.2)

A relation R on a set A is called reflexive if for all
*x* ∈*A,*(x,*x) ∈R.*

Example (7.4)

*For A*= {1,2,3,4},*a relation R*⊆*A*×*A will be reflexive if and*
*only if R* ⊇ {(1,1),(2,2),(3,3),(4,4)}.

*Consequently, R*_{1}= {(1,1),(2,2),(3,3)}is not a reflexive
*relation on A, whereas R*_{2}= {(x,*y*)|x,*y* ∈*A,x* ≤*y}*is reflexive
on A.

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.2)

A relation R on a set A is called reflexive if for all
*x* ∈*A,*(x,*x) ∈R.*

Example (7.4)

*For A*= {1,2,3,4},*a relation R*⊆*A*×*A will be reflexive if and*
*only if R* ⊇ {(1,1),(2,2),(3,3),(4,4)}.

*Consequently, R*_{1}= {(1,1),(2,2),(3,3)}is not a reflexive
*relation on A, whereas R*_{2}= {(x,*y*)|x,*y* ∈*A,x* ≤*y}*is reflexive
on A.

Definition (7.3)

Relation R on set A is called symmetric if
(x,*y) ∈R*⇒ (y,*x*) ∈*R,for all x*,*y* ∈*A.*

Example (7.6)

*With A*= {1,2,3}, we have:

a) *R*_{1}= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not
reflexive, relation on A;

b) *R*_{2}= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not
symmetric, relation on A;

c) *R*_{3}= {(1,1),(2,2),(3,3)}and

*R*_{4}= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A
that are both reflexive and symmetric; and

d) *R*_{5}= {(1,1),(2,2),(3,3)}, a relation on A that is neither
reflexive nor symmetric.

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.3)

Relation R on set A is called symmetric if
(x,*y) ∈R*⇒ (y,*x*) ∈*R,for all x*,*y* ∈*A.*

Example (7.6)

*With A*= {1,2,3}, we have:

a) *R*_{1}= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not
reflexive, relation on A;

b) *R*_{2}= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not
symmetric, relation on A;

c) *R*_{3}= {(1,1),(2,2),(3,3)}and

*R*_{4}= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A

Definition (7.3)

Relation R on set A is called symmetric if
(x,*y) ∈R*⇒ (y,*x*) ∈*R,for all x*,*y* ∈*A.*

Example (7.6)

*With A*= {1,2,3}, we have:

a) *R*_{1}= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not
reflexive, relation on A;

b) *R*_{2}= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not
symmetric, relation on A;

c) *R*_{3}= {(1,1),(2,2),(3,3)}and

*R*_{4}= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A
that are both reflexive and symmetric; and

d) *R*_{5}= {(1,1),(2,2),(3,3)}, a relation on A that is neither
reflexive nor symmetric.

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.3)

Relation R on set A is called symmetric if
(x,*y) ∈R*⇒ (y,*x*) ∈*R,for all x*,*y* ∈*A.*

Example (7.6)

*With A*= {1,2,3}, we have:

a) *R*_{1}= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not
reflexive, relation on A;

b) *R*_{2}= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not
symmetric, relation on A;

c) *R*_{3}= {(1,1),(2,2),(3,3)}and

*R*_{4}= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A

Definition (7.3)

Relation R on set A is called symmetric if
(x,*y) ∈R*⇒ (y,*x*) ∈*R,for all x*,*y* ∈*A.*

Example (7.6)

*With A*= {1,2,3}, we have:

a) *R*_{1}= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not
reflexive, relation on A;

b) *R*_{2}= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not
symmetric, relation on A;

c) *R*_{3}= {(1,1),(2,2),(3,3)}and

*R*_{4}= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A
that are both reflexive and symmetric; and

d) *R*_{5}= {(1,1),(2,2),(3,3)}, a relation on A that is neither
reflexive nor symmetric.

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.4)

For a set A, a relation R on A is called transitive if, for all
*x*,*y*,*z* ∈*A,*(x,*y*),(y,*z) ∈R*⇒ (x,*z*) ∈*R.*

Example (7.7)

All the relations in Examples 7.1 and 7.2 are transitive, as are the relations in Examples 7.3(c)

Definition (7.4)

For a set A, a relation R on A is called transitive if, for all
*x*,*y*,*z* ∈*A,*(x,*y*),(y,*z) ∈R*⇒ (x,*z*) ∈*R.*

Example (7.7)

All the relations in Examples 7.1 and 7.2 are transitive, as are the relations in Examples 7.3(c)

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.5)

Given a relation R on a set A , R is called antisymmetric if for all
*a,b*∈*A, (aRb and bRa)*⇒*a*=*b.*

Example (7.11)

*For a universe U, define the relation R on P*(U)by(A,*B) ∈R if*
*A*⊆*B, for*(A,*B) ⊆U.*

*So R is the subset relation of Chapter 3 and if ARB and BRA,*
*then we have A*⊆*B and B*⊆*A, which gives us A*=*B.*

Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.

Definition (7.5)

Given a relation R on a set A , R is called antisymmetric if for all
*a,b*∈*A, (aRb and bRa)*⇒*a*=*b.*

Example (7.11)

*For a universe U, define the relation R on P*(U)by(A,*B) ∈R if*
*A*⊆*B, for*(A,*B) ⊆U.*

*So R is the subset relation of Chapter 3 and if ARB and BRA,*
*then we have A*⊆*B and B*⊆*A, which gives us A*=*B.*

Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.5)

Given a relation R on a set A , R is called antisymmetric if for all
*a,b*∈*A, (aRb and bRa)*⇒*a*=*b.*

Example (7.11)

*For a universe U, define the relation R on P*(U)by(A,*B) ∈R if*
*A*⊆*B, for*(A,*B) ⊆U.*

*So R is the subset relation of Chapter 3 and if ARB and BRA,*
*then we have A*⊆*B and B*⊆*A, which gives us A*=*B.*

Consequently, this relation is antisymmetric, as well as reflexive and transitive, but it is not symmetric.

Definition (7.5)

Given a relation R on a set A , R is called antisymmetric if for all
*a,b*∈*A, (aRb and bRa)*⇒*a*=*b.*

Example (7.11)

*For a universe U, define the relation R on P*(U)by(A,*B) ∈R if*
*A*⊆*B, for*(A,*B) ⊆U.*

*So R is the subset relation of Chapter 3 and if ARB and BRA,*
*then we have A*⊆*B and B*⊆*A, which gives us A*=*B.*

WEN-CHINGLIEN **Discrete Mathematics**

Definition (7.6)

A relation R on a set A is called a partial order, or a partial ordering relation, if R is reflexive, antisymmetric, and transitive.

Definition (7.7)

An equivalent relation R on a set A is a relation that is reflexive, symmetric, and transitive.

Definition (7.6)

A relation R on a set A is called a partial order, or a partial ordering relation, if R is reflexive, antisymmetric, and transitive.

Definition (7.7)

An equivalent relation R on a set A is a relation that is reflexive, symmetric, and transitive.

WEN-CHINGLIEN **Discrete Mathematics**