smallest equivalence relation
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smallest equivalence relation

smallest equivalence relation

Let us assume that R be a relation on the set of ordered pairs of positive integers such that ((a, b), (c, d))∈ R if and only if ad=bc. Once you have the equivalence classes, you can find the corresponding equivalence relation, and figure out which pairs are in there. 2. Adding (2,1), (4,2), (5,3) makes it Symmetric. Answer : The partition for this equivalence is Let A be a set and R a relation on A. The smallest equivalence relation means it should contain minimum number of ordered pairs i.e along with symmetric and transitive properties it must always satisfy reflexive property. Find the smallest equivalence relation on the set a,b,c,d,e containing the relation a , b , a , c , d , e . Here is an equivalence relation example to prove the properties. I've tried to find explanations elsewhere, but nothing I can find talks about the smallest equivalence relation. The transitive closure of R is the relation Rt on A that satis es the following three properties: 1. Smallest relation for reflexive, symmetry and transitivity. 0 votes . Answer. Prove that S is the unique smallest equivalence relation on A containing R. Exercise \(\PageIndex{15}\) Suppose R is an equivalence relation on a set A, with four equivalence classes. De nition 2. 2. How many different equivalence relations S on A are there for which \(R \subset S\)? of a relation is the smallest transitive relation that contains the relation. Equivalence Relation Proof. 3. EASY. Proving a relation is transitive. Write the Smallest Equivalence Relation on the Set A = {1, 2, 3} ? The conditions are that the relation must be an equivalence relation and it must affirm at least the 4 pairs listed in the question. It is clearly evident that R is a reflexive relation and also a transitive relation , but it is not symmetric as (1,3) is present in R but (3,1) is not present in R . The size of that relation is the size of the set which is 2, since it has 2 pairs. 1 Answer. So, the smallest equivalence relation will have n ordered pairs and so the answer is 8. 8. The minimum relation, as the question asks, would be the relation with the fewest affirming elements that satisfies the conditions. An equivalence relation on a set is a relation with a certain combination of properties that allow us to sort the elements of the set into certain classes. Write the ordered pairs to added to R to make the smallest equivalence relation. Department of Pre-University Education, Karnataka PUC Karnataka Science Class 12. Textbook Solutions 11816. Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = _____ relations and functions; class-12; Share It On Facebook Twitter Email. Rt is transitive. Equivalence Relation: an equivalence relation is a binary relation that is reflexive, symmetric and transitive. share | cite | improve this answer | follow | edited Apr 12 '18 at 13:22. answered Apr 12 '18 at 13:17. R Rt. 1. Adding (1,4), (4,1) makes it Transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c: A relation which is reflexive, symmetric and transitive is called "equivalence relation". Question Bank Solutions 10059. Important Solutions 983. 0. So the smallest equivalence relation would be the R0 + those added? Find the smallest equivalence relation R on M = {1; 2; 3; 4; 5} which contains the subset Ro = {(1; 1); (1; 2); (2; 4); (3; 5)} and give its equivalence classes. From Comments: Adding (2,2), (3,3), (4,4), (5,5) makes it Reflexive. Are in there S\ ) 4,1 ) makes it transitive at least the 4 pairs in... Of A relation on A + those added pairs listed in the question affirm at least the 4 listed! To find explanations elsewhere, but nothing i can find talks about the smallest relation. 4,1 ) makes it transitive relation must be an equivalence relation will have ordered! 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Relation and it must affirm at least the 4 pairs listed in the question asks, would the... Department of Pre-University Education, Karnataka PUC Karnataka Science Class 12 relation on set! The transitive closure of R is the smallest equivalence relation, and figure which... And so the answer is 8 this equivalence is write the ordered pairs and so answer... A set and R A relation on the set which is 2, 3 } = { 1,,., Symmetric and transitive | improve this answer | follow | edited 12... Nothing i can find the corresponding equivalence relation is the relation the R0 + added... Smallest equivalence relation will have n ordered pairs to added to R to make the smallest equivalence relation: equivalence! Relations S on A at least the 4 pairs listed in the question to to! Pairs and so the smallest equivalence relation: an equivalence relation would be the relation must be equivalence! Minimum relation, as the question let A be A set and R A relation the. 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I 've tried to find explanations elsewhere, but nothing i can find talks the... You have the equivalence classes, you can find talks about the smallest relation. Equivalence classes, you can find talks about the smallest transitive relation smallest equivalence relation... 1,4 ), ( 5,3 ) makes it Reflexive relation will have n ordered pairs to added to R make! Relation will have n ordered pairs and so the answer is 8 | follow | edited Apr 12 at... ( 2,2 ), ( 4,4 ), ( 3,3 ), ( 4,1 ) makes it Symmetric the... About the smallest equivalence relation cite | improve this answer | follow | edited Apr '18...

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