Let A be a set and R a relation on A. of a relation is the smallest transitive relation that contains the relation. How many different equivalence relations S on A are there for which \(R \subset S\)? 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. EASY. Important Solutions 983. 1 Answer. From Comments: Adding (2,2), (3,3), (4,4), (5,5) makes it Reflexive. R Rt. Answer. 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 . De nition 2. Smallest relation for reflexive, symmetry and transitivity. Equivalence Relation Proof. 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. Textbook Solutions 11816. 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". So the smallest equivalence relation would be the R0 + those added? 0. Department of Pre-University Education, Karnataka PUC Karnataka Science Class 12. 2. I've tried to find explanations elsewhere, but nothing I can find talks about the smallest equivalence relation. 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. 1. Answer : The partition for this equivalence is Rt is transitive. 0 votes . Equivalence Relation: an equivalence relation is a binary relation that is reflexive, symmetric and transitive. Proving a relation is transitive. The minimum relation, as the question asks, would be the relation with the fewest affirming elements that satisfies the conditions. 2. Write the ordered pairs to added to R to make the smallest equivalence relation. Find the smallest equivalence relation on the set a,b,c,d,e containing the relation a , b , a , c , d , e . 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. Adding (1,4), (4,1) makes it Transitive. Adding (2,1), (4,2), (5,3) makes it Symmetric. 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. 3. The transitive closure of R is the relation Rt on A that satis es the following three properties: 1. 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. Write the Smallest Equivalence Relation on the Set A = {1, 2, 3} ? 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