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# Discrete Mathematics

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### Discrete Mathematics

WEN-CHING LIEN Department of Mathematics National Cheng Kung University

2008

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### 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

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### 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.

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Example (7.1)

a) Define the relation R on the set Z by aRb, or(a,b) ∈R,if ab.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 nZ+.For x,yZ,the modulo n relation R is defined by xRy if xy 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

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Example (7.1)

a) Define the relation R on the set Z by aRb, or(a,b) ∈R,if ab.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 nZ+.For x,yZ,the modulo n relation R is defined by xRy if xy 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

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Example (7.1)

a) Define the relation R on the set Z by aRb, or(a,b) ∈R,if ab.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 nZ+.For x,yZ,the modulo n relation R is defined by xRy if xy 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

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Example (7.1)

a) Define the relation R on the set Z by aRb, or(a,b) ∈R,if ab.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 nZ+.For x,yZ,the modulo n relation R is defined by xRy if xy 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

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Example (7.1 continued)

c) For the universe U = {1,2,3,4,5,6,7}consider the (fixed) set CU where C = {1,2,3,6}.Define the relation R on P(U)by ARB when AC=BC.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 XC = ∅ =YC.However, the sets S = {1,2,3,4,5}

and T = {1,2,3,6,7}are not related. Since SC= {1,2,3} 6= {1,2,3,6} =TC.

WEN-CHINGLIEN Discrete Mathematics

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Example (7.1 continued)

c) For the universe U = {1,2,3,4,5,6,7}consider the (fixed) set CU where C = {1,2,3,6}.Define the relation R on P(U)by ARB when AC=BC.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 XC = ∅ =YC.However, the sets S = {1,2,3,4,5}

and T = {1,2,3,6,7}are not related. Since SC= {1,2,3} 6= {1,2,3,6} =TC.

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Example (7.1 continued)

c) For the universe U = {1,2,3,4,5,6,7}consider the (fixed) set CU where C = {1,2,3,6}.Define the relation R on P(U)by ARB when AC=BC.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 XC = ∅ =YC.However, the sets S = {1,2,3,4,5}

and T = {1,2,3,6,7}are not related. Since SC= {1,2,3} 6= {1,2,3,6} =TC.

WEN-CHINGLIEN Discrete Mathematics

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Example (7.1 continued)

c) For the universe U = {1,2,3,4,5,6,7}consider the (fixed) set CU where C = {1,2,3,6}.Define the relation R on P(U)by ARB when AC=BC.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 XC = ∅ =YC.However, the sets S = {1,2,3,4,5}

and T = {1,2,3,6,7}are not related. Since SC= {1,2,3} 6= {1,2,3,6} =TC.

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Example (7.1 continued)

c) For the universe U = {1,2,3,4,5,6,7}consider the (fixed) set CU where C = {1,2,3,6}.Define the relation R on P(U)by ARB when AC=BC.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 XC = ∅ =YC.However, the sets S = {1,2,3,4,5}

and T = {1,2,3,6,7}are not related. Since SC= {1,2,3} 6= {1,2,3,6} =TC.

WEN-CHINGLIEN Discrete Mathematics

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Definition (7.2)

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

Example (7.4)

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

Consequently, R1= {(1,1),(2,2),(3,3)}is not a reflexive relation on A, whereas R2= {(x,y)|x,yA,xy}is reflexive on A.

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Definition (7.2)

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

Example (7.4)

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

Consequently, R1= {(1,1),(2,2),(3,3)}is not a reflexive relation on A, whereas R2= {(x,y)|x,yA,xy}is reflexive on A.

WEN-CHINGLIEN Discrete Mathematics

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Definition (7.2)

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

Example (7.4)

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

Consequently, R1= {(1,1),(2,2),(3,3)}is not a reflexive relation on A, whereas R2= {(x,y)|x,yA,xy}is reflexive on A.

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Definition (7.3)

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

Example (7.6)

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

a) R1= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not reflexive, relation on A;

b) R2= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not symmetric, relation on A;

c) R3= {(1,1),(2,2),(3,3)}and

R4= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A that are both reflexive and symmetric; and

d) R5= {(1,1),(2,2),(3,3)}, a relation on A that is neither reflexive nor symmetric.

WEN-CHINGLIEN Discrete Mathematics

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Definition (7.3)

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

Example (7.6)

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

a) R1= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not reflexive, relation on A;

b) R2= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not symmetric, relation on A;

c) R3= {(1,1),(2,2),(3,3)}and

R4= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A

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Definition (7.3)

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

Example (7.6)

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

a) R1= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not reflexive, relation on A;

b) R2= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not symmetric, relation on A;

c) R3= {(1,1),(2,2),(3,3)}and

R4= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A that are both reflexive and symmetric; and

d) R5= {(1,1),(2,2),(3,3)}, a relation on A that is neither reflexive nor symmetric.

WEN-CHINGLIEN Discrete Mathematics

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Definition (7.3)

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

Example (7.6)

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

a) R1= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not reflexive, relation on A;

b) R2= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not symmetric, relation on A;

c) R3= {(1,1),(2,2),(3,3)}and

R4= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A

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Definition (7.3)

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

Example (7.6)

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

a) R1= {(1,2),(2,1),(1,3),(3,1)} a symmetric, but not reflexive, relation on A;

b) R2= {(1,1),(2,2),(3,3),(2,3)} a reflexive, but not symmetric, relation on A;

c) R3= {(1,1),(2,2),(3,3)}and

R4= {(1,1),(2,2),(3,3),(2,3),(3,2)}, two relations on A that are both reflexive and symmetric; and

d) R5= {(1,1),(2,2),(3,3)}, a relation on A that is neither reflexive nor symmetric.

WEN-CHINGLIEN Discrete Mathematics

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Definition (7.4)

For a set A, a relation R on A is called transitive if, for all x,y,zA,(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)

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Definition (7.4)

For a set A, a relation R on A is called transitive if, for all x,y,zA,(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

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Definition (7.5)

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

Example (7.11)

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

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

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

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Definition (7.5)

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

Example (7.11)

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

So R is the subset relation of Chapter 3 and if ARB and BRA, then we have AB and BA, 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

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Definition (7.5)

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

Example (7.11)

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

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

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

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Definition (7.5)

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

Example (7.11)

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

So R is the subset relation of Chapter 3 and if ARB and BRA, then we have AB and BA, 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

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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.

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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

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### Thank you.

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung