smallest equivalence relation

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

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