Discrete Mathematics

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.

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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.

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

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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

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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 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.

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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.

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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.

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

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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, R₁= {(1,1),(2,2),(3,3)}is not a reflexive relation on A, whereas R₂= {(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) 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.

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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,y ∈A.

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

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) 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.

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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,y ∈A.

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

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) 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.

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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)

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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.

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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.

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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.

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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Thank you.