Equivalence Relation & Partitions Equivalence
Description: Equivalence Relation Partitions Equivalence Relations Partitions Definition(Different Types of Relations): Let A be a non-empty set and ρ be a relation on A . Then ρ is said to be a a)Reflexive Relation if (a,a) ϵ ρ i.e.,aρa,for each
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slide1. Equivalence Relation & Partitions Equivalence Relations & Partitions<br>
slide2. Definition(Different Types of Relations):
Let A be a non-empty set and ρ be a relation on A . Then ρ is said to be a
a)Reflexive Relation if (a,a) ϵ ρ i.e.,aρa,for each aϵA ;
b)Symmetric Relation if (a,b) ϵ ρ implies that (b,a) ϵ ρ i.e., for any two elements a,b ϵA,whenever
aρb holds then bρa holds;
c)Transitive Relation if for any three elements a,b,c ϵ A,whenever aρb and bρc hold then aρc holds<br>
slide4. Equivalance Relation
The relation ρ is said to be equivalance if it is reflexive,symmetric & transitive.<br>
slide5. Examples:
Question 1:Let us assume that ρ is a relation on the set Z of integers
defined by xρy if and only if x-y is an integer. Prove that ρ is an equivalence relation on Z.
Solution:Reflexive: Consider x ϵ Z,then x – x = 0 which is an integer. Therefore xρx.
Symmetric: Consider x , y ϵ Z and xρy.Then x – y is an integer. Thus, y – x = – ( x – y), y – x is also an integer. Therefore yρx.
Transitive: Consider x , y ϵ Z , xρy and yρz. Therefore x-y and y-z are integers. According to the transitive property, ( x – y ) + ( y – z ) = x – z is also an integer. So that xρz.
Thus, ρ is an equivalence relation on Z.<br>
slide2. Definition(Different Types of Relations):
Let A be a non-empty set and ρ be a relation on A . Then ρ is said to be a
a)Reflexive Relation if (a,a) ϵ ρ i.e.,aρa,for each aϵA ;
b)Symmetric Relation if (a,b) ϵ ρ implies that (b,a) ϵ ρ i.e., for any two elements a,b ϵA,whenever
aρb holds then bρa holds;
c)Transitive Relation if for any three elements a,b,c ϵ A,whenever aρb and bρc hold then aρc holds<br>
slide4. Equivalance Relation
The relation ρ is said to be equivalance if it is reflexive,symmetric & transitive.<br>
slide5. Examples:
Question 1:Let us assume that ρ is a relation on the set Z of integers
defined by xρy if and only if x-y is an integer. Prove that ρ is an equivalence relation on Z.
Solution:Reflexive: Consider x ϵ Z,then x – x = 0 which is an integer. Therefore xρx.
Symmetric: Consider x , y ϵ Z and xρy.Then x – y is an integer. Thus, y – x = – ( x – y), y – x is also an integer. Therefore yρx.
Transitive: Consider x , y ϵ Z , xρy and yρz. Therefore x-y and y-z are integers. According to the transitive property, ( x – y ) + ( y – z ) = x – z is also an integer. So that xρz.
Thus, ρ is an equivalence relation on Z.<br>