Mathematics | Closure of Relations and Equivalence Relations

13 Dec, 2019
Prerequisite : Introduction to Relations, Representation of Relations
Combining Relations :
As we know that relations are just sets of ordered pairs, so all set operations apply to them as well. Two relations can be combined in several ways such as –
Union –  consists of all ordered pairs from both relations. Duplicate ordered pairs removed from Union.
Intersection –  consists of ordered pairs which are in both relations.
Difference –  consists of all ordered pairs only in , but not in .
Symmetric Difference –  consists of all ordered pairs which are either in  or  but not both.
There is another way two relations can be combined that is analogous to the composition of functions.
Composition – Let  be a relation from  to  and  be a relation from  to , then the composite of  and , denoted by , is the relation consisting of ordered pairs  where  and for which there exists an element  such that  and .
Example – What is the composite of the relations  and  where  is a relation from  to  with  and  is a relation from  to  with ?
Solution – By computing all ordered pairs where the first element belongs to  and the second element belongs to , we get –
Composition of Relation on itself :
A relation can be composed with itself to obtain a degree of separation between the elements of the set on which  is defined.
Let  be a relation on the set .
The powers  where  are defined recursively by -
 and .

Theorem – Let  be a relation on set A, represented by a di-graph. There is a path of length , where  is a positive integer, from  to  if and only if .
Important Note : A relation  on set  is transitive if and only if  for 
Closure of Relations :
Consider a relation  on set .  may or may not have a property , such as reflexivity, symmetry, or transitivity.
If there is a relation  with property  containing  such that  is the subsetof every relation with property  containing , then  is called the closure of with respect to .
We can obtain closures of relations with respect to property  in the following ways –
Reflexive Closure –  is the diagonal relation on set . The reflexive closure of relation  on set  is .Symmetric Closure – Let  be a relation on set , and let  be the inverse of . The symmetric closure of relation  on set  is .Transitive Closure – Let  be a relation on set . The connectivity relation is defined as – . The transitive closure of  is .
Reflexive Closure –  is the diagonal relation on set . The reflexive closure of relation  on set  is .
Symmetric Closure – Let  be a relation on set , and let  be the inverse of . The symmetric closure of relation  on set  is .
Transitive Closure – Let  be a relation on set . The connectivity relation is defined as – . The transitive closure of  is .
Example – Let  be a relation on set  with . Find the reflexive, symmetric, and transitive closure of R.
Solution –For the given set, . So the reflexive closure of  is 
For the symmetric closure we need the inverse of , which is.The symmetric closure of  is-
For the transitive closure, we need to find . we need to find  until . We stop when this condition is achieved since finding higher powers of  would be the same.Since,  we stop the process.Transitive closure,  –
Equivalence Relations :
Let  be a relation on set . If  is reflexive, symmetric, and transitive then it is said to be a equivalence relation.Consequently, two elements  and  related by an equivalence relation are said to be equivalent.
Example – Show that the relation is an equivalence relation.  is the congruence modulo  function. It is true if and only if  divides .
Solution – To show that the relation is an equivalence relation we must prove that the relation is reflexive, symmetric and transitive.
Reflexive – For any element ,  is divisible by .. So, congruence modulo  is reflexive.Symmetric – For any two elements  and , if  or  i.e.  is divisible by , then  is also divisible by .. So Congruence Modulo  is symmetric.Transitive – For any three elements , , and  if  then-Adding both equations,    . So,  is transitive.Since the relation  is reflexive, symmetric, and transitive, we conclude that  is an equivalence relation.Equivalence Classes :Let  be an equivalence relation on set .We know that if  then  and  are said to be equivalent with respect to .The set of all elements that are related to an element  of  is called theequivalence class of . It is denoted by  or simply  if there is only onerelation to consider.Formally,Any element  is said to be the representative of .Important Note : All the equivalence classes of a Relation  on set  are either equal or disjoint and their union gives the set .The equivalence classes are also called partitions since they are disjoint and their union gives the set on which the relation is definedExample : What are the equivalence classes of the relation Congruence Modulo ?Solution : Let  and  be two numbers such that . This means that the remainder obtained by dividing  and  with  is the same.Possible values for the remainder- Therefore, there are  equivalence classes – GATE CS Corner QuestionsPracticing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them.1. GATE CS 2013, Question 12. GATE CS 2005, Question 423. GATE CS 2001, Question 24. GATE CS 2000, Question 28References –Composition of Relations – WikipediaDiscrete Mathematics and its Applications, by Kenneth H RosenThis article is contributed by Chirag Manwani. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.My Personal Notes
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Reflexive – For any element ,  is divisible by .. So, congruence modulo  is reflexive.
Symmetric – For any two elements  and , if  or  i.e.  is divisible by , then  is also divisible by .. So Congruence Modulo  is symmetric.
Transitive – For any three elements , , and  if  then-Adding both equations,    . So,  is transitive.
    
. So,  is transitive.
Since the relation  is reflexive, symmetric, and transitive, we conclude that  is an equivalence relation.
Equivalence Classes :
Let  be an equivalence relation on set .We know that if  then  and  are said to be equivalent with respect to .
The set of all elements that are related to an element  of  is called theequivalence class of . It is denoted by  or simply  if there is only onerelation to consider.Formally,
Any element  is said to be the representative of .Important Note : All the equivalence classes of a Relation  on set  are either equal or disjoint and their union gives the set .The equivalence classes are also called partitions since they are disjoint and their union gives the set on which the relation is defined
Example : What are the equivalence classes of the relation Congruence Modulo ?
Solution : Let  and  be two numbers such that . This means that the remainder obtained by dividing  and  with  is the same.Possible values for the remainder- Therefore, there are  equivalence classes – 
GATE CS Corner Questions
Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them.
1. GATE CS 2013, Question 12. GATE CS 2005, Question 423. GATE CS 2001, Question 24. GATE CS 2000, Question 28
References –Composition of Relations – WikipediaDiscrete Mathematics and its Applications, by Kenneth H Rosen
This article is contributed by Chirag Manwani. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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