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