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