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". Here is an equivalence relation example to prove the properties. of a relation is the smallest transitive relation that contains the relation. Answer : The partition for this equivalence is 2. Important Solutions 983. So the smallest equivalence relation would be the R0 + those added? The conditions are that the relation must be an equivalence relation and it must affirm at least the 4 pairs listed in the question. De nition 2. Smallest relation for reflexive, symmetry and transitivity. 1 Answer. Let A be a set and R a relation on A. The transitive closure of R is the relation Rt on A that satis es the following three properties: 1. I've tried to find explanations elsewhere, but nothing I can find talks about the smallest equivalence relation. R Rt. EASY. Once you have the equivalence classes, you can find the corresponding equivalence relation, and figure out which pairs are in there. Find the smallest equivalence relation on the set a,b,c,d,e containing the relation a , b , a , c , d , e . From Comments: Adding (2,2), (3,3), (4,4), (5,5) makes it Reflexive. 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 . 0 votes . 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. Equivalence Relation: an equivalence relation is a binary relation that is reflexive, symmetric and transitive. Write the ordered pairs to added to R to make the smallest equivalence relation. 0. Equivalence Relation Proof. The size of that relation is the size of the set which is 2, since it has 2 pairs. Let us assume that R be a relation on the set of ordered pairs of positive integers such that ((a, b), (c, d))â R if and only if ad=bc. 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. Adding (2,1), (4,2), (5,3) makes it Symmetric. share | cite | improve this answer | follow | edited Apr 12 '18 at 13:22. answered Apr 12 '18 at 13:17. How many different equivalence relations S on A are there for which \(R \subset S\)? Department of Pre-University Education, Karnataka PUC Karnataka Science Class 12. 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. Proving a relation is transitive. 3. The minimum relation, as the question asks, would be the relation with the fewest affirming elements that satisfies the conditions. Adding (1,4), (4,1) makes it Transitive. Write the Smallest Equivalence Relation on the Set A = {1, 2, 3} ? Textbook Solutions 11816. 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. 8. 1. Rt is transitive. 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. 2. Question Bank Solutions 10059. So, the smallest equivalence relation will have n ordered pairs and so the answer is 8. Answer. 4 pairs listed in the question cite | improve this smallest equivalence relation | follow | Apr. It must affirm at least the 4 pairs listed in the question asks, would be the R0 those..., since it has 2 pairs Apr 12 '18 at 13:17 the corresponding relation! Are that the relation must be an equivalence relation on A that satis es the following three:! 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