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. De nition 2. So, the smallest equivalence relation will have n ordered pairs and so the answer is 8. 2. EASY. Department of Pre-University Education, Karnataka PUC Karnataka Science Class 12. Equivalence Relation: an equivalence relation is a binary relation that is reflexive, symmetric and transitive. 2. share | cite | improve this answer | follow | edited Apr 12 '18 at 13:22. answered Apr 12 '18 at 13:17. 1 Answer. Textbook Solutions 11816. of a relation is the smallest transitive relation that contains the relation. The conditions are that the relation must be an equivalence relation and it must affirm at least the 4 pairs listed in the question. 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 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". The size of that relation is the size of the set which is 2, since it has 2 pairs. R Rt. 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. 0 votes . The transitive closure of R is the relation Rt on A that satis es the following three properties: 1. Answer : The partition for this equivalence is 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 . 3. From Comments: Adding (2,2), (3,3), (4,4), (5,5) makes it Reflexive. 1. 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. Smallest relation for reflexive, symmetry and transitivity. The minimum relation, as the question asks, would be the relation with the fewest affirming elements that satisfies the conditions. Rt is transitive. Answer. Equivalence Relation Proof. 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. Question Bank Solutions 10059. Write the Smallest Equivalence Relation on the Set A = {1, 2, 3} ? So the smallest equivalence relation would be the R0 + those added? 8. 0. Write the ordered pairs to added to R to make the smallest equivalence relation. 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