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1. Dictionary

   E·quiv·a·lence re·la·tion

   /əˈkwivələns/

   noun

   • 1. a relation between elements of a set that is reflexive, symmetric, and transitive. It thus defines exclusive classes whose members bear the relation to each other and not to those in other classes (e.g., “having the same value of a measured property”).

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2. • Reflexive, symmetric, and transitive

     • A relation R on a set is called an equivalence relation if and only if the relation is reflexive, symmetric, and transitive. The equivalence relation is a relationship on the set which is generally represented by the symbol “∼”. Equivalence Relation Definition
     www.geeksforgeeks.org › equivalence-relations

     Equivalence Relations: Definition, Proof, Properties, Examples

3. Top results related to define equivalence relation

   • Equivalence Relation on a Set

     A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on. 

     What is an Equivalence Relation?A relation R on a set A is called an equivalence relation if it is

     • Reflexive Relation: (a, a) ∈ R ∀ a ∈ A, i.e. aRa for all a ∈ A.
     • Symmetric Relation: ∀ a, b ∈ A, (a, b) ∈ R ↔ (b, a) ∈ R.
     • Transitive Relation: ∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.

     where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

     Example: Consider set A = {a, b}

     R = {(a, a), (a, b), (b, a)} is not equivalence as for (b, b) tuple is not present but R = {(a, a), (a, b), (b, a), (b, b)} is an equivalence relation.

     How to verify an Equivalence Relation?The process of identifying/verifying if any given relation is equivalency:

     • Check if the relation is Reflexive.
     • Check if the relation is Symmetric.
     • Check if the relation is Transitive.

     Follow the below illustration for a better understanding

     Example: Consider set R = {(1, 1), (1, 3), (2, 2), (3, 3), (3, 1), (3, 2), (3, 4), (4, 4), (4, 3)}

     Pairs (1, 1), (2, 2), (3, 3), (4, 4) exist:   ⇒ This satisfies the reflexive condition.

     The symmetric condition is also satisfied.

     For the pairs (1, 3) and (3, 4):    ⇒ The...

     Check all answers on Geeksforgeeks

     1/5

   • Equivalence Relation on a Set

     A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on. 

     What is an Equivalence Relation?

     A relation R on a set A is called an equivalence relation if it is

     • Reflexive Relation: (a, a) ∈ R ∀ a ∈ A, i.e. aRa for all a ∈ A.
     • Symmetric Relation: ∀ a, b ∈ A, (a, b) ∈ R ↔ (b, a) ∈ R.
     • Transitive Relation: ∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.

     where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

     Example: Consider set A = {a, b}...

     Check all answers on Geeksforgeeks

     2/5

   • Equivalence Relation on a Set

     A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on. 

     What is an Equivalence Relation?A relation R on a set A is called an equivalence relation if it is

     • Reflexive Relation: (a, a) ∈ R ∀ a ∈ A, i.e. aRa for all a ∈ A.
     • Symmetric Relation: ∀ a, b ∈ A, (a, b) ∈ R ↔ (b, a) ∈ R.
     • Transitive Relation: ∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.

     where R is a subset of (A x A), i.e. the cartesian product of set A with itself.

     Example: Consider set A = {a, b}

     R = {(a, a), (a, b), (b, a)} is not equivalence as for (b, b) tuple is not present but R = {(a, a), (a, b), (b, a), (b, b)} is an equivalence relation.

     How to verify an Equivalence Relation?The process of identifying/verifying if any given relation is equivalency:

     • Check if the relation is Reflexive.
     • Check if the relation is Symmetric.
     • Check if the relation is Transitive.

     Follow the below illustration for a better understanding

     Example: Consider set R = {(1, 1), (1, 3), (2, 2), (3, 3), (3, 1), (3, 2), (3, 4), (4, 4), (4, 3)}

     Pairs (1, 1), (2, 2), (3, 3), (4, 4) exist:   ⇒ This satisfies the reflexive condition.

     The symmetric condition is also satisfied.

     For the pairs (1, 3) and (3, 4):    ⇒ The...

     Check all answers on Geeksforgeeks

     3/5

   • Define a transitive relation over facts

     Top Answer

     Answered Dec 18, 2015 · 2 votes

     You can perhaps use big_fact/2 to describe the database:

     big_fact(elephant,horse).big_fact(horse,ant).big_fact(ant, bacteria).big_fact(bacteria,virus).-

     Then you can use bigger/2 as a transitive closure:

     bigger(X,Y) :- big_fact(X,Y).bigger(X,Z) :- big_fact(X,Y), bigger(Y,Z).

     If you run this in swi-prolog, it generates:

     ?- bigger(X,Y).X = elephant,Y = horse ;X = horse,Y = ant ;X = ant,Y = bacteria ;X = bacteria,Y = virus ;X = elephant,Y = ant ;X = elephant,Y = bacteria ;X = elephant,Y = virus ;X = horse,Y = bacteria ;X = horse,Y = virus ;X = ant,Y = virus ;false.?- bigger(virus,Y).false.?- bigger(ant,Y).Y = bacteria ;Y = virus ;false.?- bigger(X,horse).X = elephant ;false.?- bigger(X,bacteria).X = ant ;X = elephant ;X = horse ;false.?- bigger(elephant,bacteria).true ;false.

     So it is a transitive closure.

     You can then define smaller/2 by means of bigger/2:

     smaller(X,Y) :- bigger(Y,X).-

     Check all answers on Stack Overflow

     4/5

   • Equivalence between regular expressions

     Top Answer

     Answered Mar 03, 2019 · 0 votes

     (2) requires a lot of rewrite and equivalence rules for you to guarantee this in the general case. Of course, if you're just trying to do this for these two specific regex's, then you can work ad hoc.

     (1) is, however, the "real" way to do it. I'm surprised you know how to use Arden's lemma, but not the reverse direction, which is far more common (in textbooks and practice) and easier. Literally any formal language book or any compiler book will give you at least one algorithm to map a regex to a DFA and also to minimize a DFA. In the event you don't have access to such books, I point you to two article I wrote ages ago, each presenting a taxonomy (at that time comprehensive) of algorithms, one for mapping a regex to a DFA and the other for minimising a DFA: http://www.kornai.com/EFS/OnlineSupportMaterial/Watson/Papers/constax.600dpi.ps https://www.researchgate.net/publication/2247379_A_Taxonomy_of_Finite_Automata_Minimization_Algorithms

     Lastly, more comprehensive work is in my own phd called "Taxonomies and Toolkits of Regular Language Algorithms", from 1995.

     BTW, I should also mention there are many toolkits implementing these algorithms, in a variety of languages.

     Best regards, Bruce

     Check all answers on Stack Overflow

     5/5

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4. People also ask

   What is an equivalence relation on a set?

   • A relation R on a set is called an equivalence relation if and only if the relation is reflexive, symmetric, and transitive. The equivalence relation is a relationship on the set which is generally represented by the symbol “∼”. An equivalence relation on a set is a binary relation that satisfies three fundamental properties:

   Equivalence Relations: Definition, Proof, Properties, Examples

   www.geeksforgeeks.org/equivalence-relations/

   See all results for this question

   Why are equivalence relations important?

   • Equivalence relations are essential in various mathematical and theoretical contexts, including algebra, set theory, and graph theory, as they provide a structured way to compare and classify elements within a set.

   Equivalence Relations: Definition, Proof, Properties, Examples

   www.geeksforgeeks.org/equivalence-relations/

   See all results for this question

   What is an example of an equivalence relation?

   • Equivalence relations are often denoted by the symbol “≡” or by writing “∼” between related elements. An example of an equivalence relation is the “congruence modulo n” relation in modular arithmetic, where two integers are related if their difference is a multiple of n. This relation is reflexive, symmetric, and transitive.

   Equivalence Relations: Definition, Proof, Properties, Examples

   www.geeksforgeeks.org/equivalence-relations/

   See all results for this question

   What properties make a relation an equivalence relation?

   • The properties that make a relation an equivalence relation (reflexivity, symmetry, and transitivity) are designed to ensure that equivalence classes exist and do provide us with the desired partition. For the beginning proof writer this all may seem very complicated, but take heart!

   6.3: Equivalence Relations - Mathematics LibreTexts

   math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06:_Relations_and_Functions/6.03:_Equivalence_Relations

   See all results for this question

5. www.geeksforgeeks.org › equivalence-relationsEquivalence Relations: Definition, Proof, Properties, Examples

   www.geeksforgeeks.org › equivalence-relations

   • Cached

   Nov 9, 2023 · Define Equivalence Relation. An equivalence relation is a binary relation on a set that satisfies three properties: reflexivity, symmetry, and transitivity. It is a way to partition a set into distinct subsets or “ equivalence classes.

6. www.cuemath.com › algebra › equivalence-relationsEquivalence Relation - Definition, Proof, Properties, Examples

   www.cuemath.com › algebra › equivalence-relations

   • Cached

   An equivalence relation is a binary relation defined on a set X such that the relation is reflexive, symmetric and transitive. The equivalence relation divides the set into disjoint equivalence classes.

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8. math.libretexts.org › Bookshelves › Mathematical7.2: Equivalence Relations - Mathematics LibreTexts

   math.libretexts.org › Bookshelves › Mathematical

   • Cached

   Apr 17, 2022 · 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. In this section, we will focus on the properties that define an equivalence relation, and in the next section, we will see how these properties allow us to sort or partition the elements of ...

9. math.libretexts.org › Bookshelves › Combinatorics_and7.3: Equivalence Relations - Mathematics LibreTexts

   math.libretexts.org › Bookshelves › Combinatorics_and

   • Cached

   A relation on a set A is an equivalence relation if it is reflexive, symmetric, and transitive. We often use the tilde notation a∼b to denote an equivalence relation.

10. math.libretexts.org › Bookshelves › Mathematical6.3: Equivalence Relations - Mathematics LibreTexts

    math.libretexts.org › Bookshelves › Mathematical

    • Cached

    Definition: Equivalence Relation. A relation \(\text{R}\) on a set \(S\) is an equivalence relation iff \(\text{R}\) is reflexive, symmetric and transitive.

11. Videos

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      Equivalence Relation in Discrete Mathematics with examples

      Mar 8, 2021

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      Equivalence Relation (Solved Problems)

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12. web.math.ucsb.edu › math8 › handoutsMath 8: Equivalence Relations - UC Santa Barbara

    web.math.ucsb.edu › math8 › handouts

    Proposition Let S be a set and let ∼ be an equivalence relation on S. For all a, b ∈ S, a ∈ cl(a) cl(a) = cl(b) if and only if a ∼ b. cl(a)∩ cl(b) = ∅ if and only if a 6∼b. Proof: Let a, b ∈ S be given. (i) We know that cl(a) = {s ∈ S : s ∼ a}. Since ∼ is reflexive, we know that a ∼ a. This means, by definition, that a ∈ cl(a).

13. web.math.ucsb.edu › ~padraic › mathcamp_2014Lecture 3: Equivalence Relations

    web.math.ucsb.edu › ~padraic › mathcamp_2014

    Today's concepts are the ideas of sets and equivalence relations: 1 Sets. set, for the purposes of this lecture, is just some collection of objects1. We usually denote a set by listing its elements in between a pair of curly braces fg.