Discrete Mathematics with Applications
MATH236
Dr. Hung P. Tong-Viet
School of Mathematics, Statistics and Computer Science
University of KwaZulu-Natal
Pietermaritzburg Campus
Semester 1, 2013
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Table of contents
1
Equivalence relations
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Equivalence relations
Equivalence relation
Definition (Equivalence relation)
A relation R on a set S is an equivalence relation if it is:
1
reflexive
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Equivalence relations
Equivalence relation
Definition (Equivalence relation)
A relation R on a set S is an equivalence relation if it is:
1
reflexive
2
symmetric and
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Equivalence relations
Equivalence relation
Definition (Equivalence relation)
A relation R on a set S is an equivalence relation if it is:
1
reflexive
2
symmetric and
3
transitive
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Equivalence relations
Equivalence relation
Definition (Equivalence relation)
A relation R on a set S is an equivalence relation if it is:
1
reflexive
2
symmetric and
3
transitive
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Equivalence relations
Equivalence relation (cont.)
Example
The relation R on Z defined by xRy iff 2 | x − y is an equivalence relation
As 2 | 0 = x − x for all x ∈ Z, R is reflexive
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Equivalence relations
Equivalence relation (cont.)
Example
The relation R on Z defined by xRy iff 2 | x − y is an equivalence relation
As 2 | 0 = x − x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 2 | x − y so that 2 | −(x − y ) = y − x,
hence y Rx. So, R is symmetric
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Equivalence relations
Equivalence relation (cont.)
Example
The relation R on Z defined by xRy iff 2 | x − y is an equivalence relation
As 2 | 0 = x − x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 2 | x − y so that 2 | −(x − y ) = y − x,
hence y Rx. So, R is symmetric
For x, y , z ∈ Z, assume that xRy and y Rz. Then 2 | x − y and
2 | y − z so that 2 | (x − y ) + (y − z) = x − z hence xRz. Thus R is
transitive
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Equivalence relations
Equivalence relation (cont.)
Example
The relation R on Z defined by xRy iff 2 | x − y is an equivalence relation
As 2 | 0 = x − x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 2 | x − y so that 2 | −(x − y ) = y − x,
hence y Rx. So, R is symmetric
For x, y , z ∈ Z, assume that xRy and y Rz. Then 2 | x − y and
2 | y − z so that 2 | (x − y ) + (y − z) = x − z hence xRz. Thus R is
transitive
Therefore, R is an equivalence relation on Z
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Equivalence relations
Equivalence relation (cont.)
Example
The relation R on Z defined by xRy iff 2 | x − y is an equivalence relation
As 2 | 0 = x − x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 2 | x − y so that 2 | −(x − y ) = y − x,
hence y Rx. So, R is symmetric
For x, y , z ∈ Z, assume that xRy and y Rz. Then 2 | x − y and
2 | y − z so that 2 | (x − y ) + (y − z) = x − z hence xRz. Thus R is
transitive
Therefore, R is an equivalence relation on Z
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Equivalence relations
Equivalence relation (cont.)
Example
Let R be a relation on Z defined by xRy iff 3 | x + 2y . Is R an
equivalence relation on Z?
As 3 | 3x = x + 2x for all x ∈ Z, R is reflexive
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Equivalence relations
Equivalence relation (cont.)
Example
Let R be a relation on Z defined by xRy iff 3 | x + 2y . Is R an
equivalence relation on Z?
As 3 | 3x = x + 2x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 3 | x + 2y so that
3 | 3(x + y ) − (x + 2y ) = y + 2x, hence y Rx. So, R is symmetric
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Equivalence relations
Equivalence relation (cont.)
Example
Let R be a relation on Z defined by xRy iff 3 | x + 2y . Is R an
equivalence relation on Z?
As 3 | 3x = x + 2x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 3 | x + 2y so that
3 | 3(x + y ) − (x + 2y ) = y + 2x, hence y Rx. So, R is symmetric
For x, y , z ∈ Z, assume that xRy and y Rz. Then 3 | x + 2y and
3 | y + 2z so that 3 | (x + 2y ) + (y + 2z) = (x + 2z) + 3y , hence
3 | x + 2z so xRz. Thus R is transitive
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Equivalence relations
Equivalence relation (cont.)
Example
Let R be a relation on Z defined by xRy iff 3 | x + 2y . Is R an
equivalence relation on Z?
As 3 | 3x = x + 2x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 3 | x + 2y so that
3 | 3(x + y ) − (x + 2y ) = y + 2x, hence y Rx. So, R is symmetric
For x, y , z ∈ Z, assume that xRy and y Rz. Then 3 | x + 2y and
3 | y + 2z so that 3 | (x + 2y ) + (y + 2z) = (x + 2z) + 3y , hence
3 | x + 2z so xRz. Thus R is transitive
Therefore, R is an equivalence relation on Z
Tong-Viet (UKZN)
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Equivalence relations
Equivalence relation (cont.)
Example
Let R be a relation on Z defined by xRy iff 3 | x + 2y . Is R an
equivalence relation on Z?
As 3 | 3x = x + 2x for all x ∈ Z, R is reflexive
For x, y ∈ Z, if xRy , then 3 | x + 2y so that
3 | 3(x + y ) − (x + 2y ) = y + 2x, hence y Rx. So, R is symmetric
For x, y , z ∈ Z, assume that xRy and y Rz. Then 3 | x + 2y and
3 | y + 2z so that 3 | (x + 2y ) + (y + 2z) = (x + 2z) + 3y , hence
3 | x + 2z so xRz. Thus R is transitive
Therefore, R is an equivalence relation on Z
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Equivalence relations
Equivalence relation (cont.)
Example
Let R1 be a relation on Z defined by xR1 y iff 5 | 2x + 3y . Is R1 an
equivalence relation on Z?
Let R2 be a relation on Z defined by xR2 y iff 3 | x − 2y . Is R2 an
equivalence relation on Z?
Answer
R1 is an equivalence relation (Check!)
But R2 is NOT an equivalence (Check!)
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