# equivalence class examples

The subsets form a partition $$P$$ of $$A$$ if, There is a direct link between equivalence classes and partitions. Equivalence Classes Definitions. Let be an equivalence relation on the set, and let. $\left\{ {1,2} \right\}$, The set $$B = \left\{ {1,2,3} \right\}$$ has $$5$$ partitions: We also use third-party cookies that help us analyze and understand how you use this website. This means that two equal sets will always be equivalent but the converse of the same may or may not be true. $\forall\, a \in A,a \in \left[ a \right]$, Two elements $$a, b \in A$$ are equivalent if and only if they belong to the same equivalence class. Example: Let A = {1, 2, 3}                   Clearly (R-1)-1 = R, Example2: R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (3, 2)} Notice an equivalence class is a set, so a collection of equivalence classes is a collection of sets. I'll leave the actual example below. {\left( {c,b} \right),\left( {c,c} \right)} \right\}}\], So, the relation $$R$$ in roster form is given by, ${R = \left\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {a,c} \right),}\right.}\kern0pt{\left. Equivalence Classes Definitions. Boundary value analysis is based on testing at the boundaries between partitions. The possible remainders for $$n = 3$$ are $$0,1,$$ and $$2.$$ An equivalence class consists of those integers that have the same remainder. The equivalence class of an element $$a$$ is denoted by $$\left[ a \right].$$ Thus, by definition, Click or tap a problem to see the solution. Let $$R$$ be an equivalence relation on a set $$A,$$ and let $$a \in A.$$ The equivalence class of $$a$$ is called the set of all elements of $$A$$ which are equivalent to $$a.$$. For each non-reflexive element its reverse also belongs to $$R:$$, \[{\left( {a,b} \right),\left( {b,a} \right) \in R,\;\;}\kern0pt{\left( {c,d} \right),\left( {d,c} \right) \in R,\;\; \ldots }$. With this approach, the family is dependent on the team member, if any member works well then whole family will function well. Equivalence Relation Examples. maybe this example i found can help: If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. Equivalence Class Testing: Boundary Value Analysis: 1.                  R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} A relation R on a set A is called an equivalence relation if it satisfies following three properties: Example: Let A = {1, 2, 3, 4} and R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}. The standard class representatives are taken to be 0, 1, 2,...,. A set of class representatives is a subset of which contains exactly one element from each equivalence class. Each test case is representative of a respective class.                     R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} What is Equivalence Class Testing? Then we will look into equivalence relations and equivalence classes. The relation $$R$$ is symmetric and transitive. }\) Similarly, we find pairs with the elements related to $$d$$ and $$e:$$ $${\left( {d,c} \right),}$$ $${\left( {d,d} \right),}$$ $${\left( {d,e} \right),}$$ $${\left( {e,c} \right),}$$ $${\left( {e,d} \right),}$$ and $${\left( {e,e} \right). By Sita Sreeraman; ISTQB, Software Testing (QA) Equivalence Partitioning: The word Equivalence means the condition of being equal or equivalent in value, worth, function, etc. Equivalence Partitioning is a black box technique to identify test cases systematically and is often the first … What is Equivalence Class Testing? The subsets \(\left\{{}\right\},\left\{ {0,2,1} \right\},\left\{ {4,3,5} \right\}$$ are not a partition because they have the empty set. It can be applied to any level of the software testing, designed to divide a sets of test conditions into the groups or sets that can be considered the same i.e. The equivalence classes of $$R$$ are defined by the expression $$\left\{ { – 1 – n, – 1 + n} \right\},$$ where $$n$$ is an integer. © Copyright 2011-2018 www.javatpoint.com. Relation R is Symmetric, i.e., aRb ⟹ bRa Relation R is transitive, i.e., aRb and bRc ⟹ aRc. Equivalence Class Testing is a type of black box technique. We know that each integer has an equivalence class for the equivalence relation of congruence modulo 3. Hence selecting one input from each group to design the test cases. Transcript.                     R-1 = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2)} The equivalence class of an element $$a$$ is denoted by $$\left[ a \right].$$ Thus, by definition, For any equivalence relation on a set $$A,$$ the set of all its equivalence classes is a partition of $$A.$$, The converse is also true. Given a set A with an equivalence relation R on it, we can break up all elements in A … E.g. This is because there is a possibility that the application may … Note that $$a\in [a]_R$$ since $$R$$ is reflexive. Question 1 Let A ={1, 2, 3, 4}. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. Let $$R$$ be an equivalence relation on a set $$A,$$ and let $$a \in A.$$ The equivalence class of $$a$$ is called the set of all elements of $$A$$ which are equivalent to $$a.$$. $\left\{ {1,3} \right\},\left\{ 2 \right\}$ aRa ∀ a∈A. $\left\{ {1,2} \right\},\left\{ 3 \right\}$ The equivalence class testing, is also known as equivalence class portioning, which is used to subdivide or partition into multiple groups of test inputs that are of similar behavior.                  R1∪ R2= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}. A relation R on a set A is called an equivalence relation if it satisfies following three properties: Relation R is Reflexive, i.e. Pick a single value from range 1 to 1000 as a valid test case. {\left( { – 11,9} \right),\left( { – 11, – 11} \right)} \right\}}\], As it can be seen, $${E_{2}} = {E_{- 2}},$$ $${E_{10}} = {E_{ – 10}}.$$ It follows from here that we can list all equivalence classes for $$R$$ by using non-negative integers $$n.$$. We'll assume you're ok with this, but you can opt-out if you wish. Equivalence class testing (Equivalence class Partitioning) is a black-box testing technique used in software testing as a major step in the Software development life cycle (SDLC). You also have the option to opt-out of these cookies. Let R be any relation from set A to set B. The next step from boundary value testing Motivation of Equivalence class testing Robustness Single/Multiple fault assumption. So, in Example 6.3.2, $$[S_2] =[S_3]=[S_1] =\{S_1,S_2,S_3\}.$$ This equality of equivalence classes will be formalized in Lemma 6.3.1. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Example: Let A = {1, 2, 3} Then if ~ was an equivalence relation for ‘of the same age’, one equivalence class would be the set of all 2-year-olds, and another the set of all 5-year-olds. Hence, there are $$3$$ equivalence classes in this example: $\left[ 0 \right] = \left\{ { \ldots , – 9, – 6, – 3,0,3,6,9, \ldots } \right\}$, $\left[ 1 \right] = \left\{ { \ldots , – 8, – 5, – 2,1,4,7,10, \ldots } \right\}$, $\left[ 2 \right] = \left\{ { \ldots , – 7, – 4, – 1,2,5,8,11, \ldots } \right\}$, Similarly, one can show that the relation of congruence modulo $$n$$ has $$n$$ equivalence classes $$\left[ 0 \right],\left[ 1 \right],\left[ 2 \right], \ldots ,\left[ {n – 1} \right].$$, Let $$A$$ be a set and $${A_1},{A_2}, \ldots ,{A_n}$$ be its non-empty subsets. For example, “3+3”, “half a dozen” and “number of kids in the Brady Bunch” all equal 6! It includes maximum, minimum, inside or outside boundaries, typical values and error values. If a member of set is given as an input, then one valid and one invalid equivalence class is defined. We will see how an equivalence on a set partitions the set into equivalence classes. Examples of Equivalence Classes. In equivalence partitioning, inputs to the software or system are divided into groups that are expected to exhibit similar behavior, so they are likely to be proposed in the same way. The set of all the equivalence classes is denoted by ℚ. Partitions A partition of a set S is a family F of non-empty subsets of S such that (i) if A and B are in F then either A = B or A ∩ B = ∅, and (ii) union A∈F A= S. S. Partitions … maybe this example i found can help: If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. If so, what are the equivalence classes of R? The equivalence class of an element $$a$$ is denoted by $$\left[ a \right].$$ Thus, by definition, ${\left[ a \right] = \left\{ {b \in A \mid aRb} \right\} }={ \left\{ {b \in A \mid a \sim b} \right\}.}$. Question 1: Let assume that F is a relation on the set R real numbers defined by xFy if and only if x-y is an integer.                 R-1 = {(1, 1), (2, 2), (3, 3), (2, 1), (3, 2), (2, 3)}.                  R2 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)} $$R$$ is transitive. Let R be the relation on the set A = {1,3,5,9,11,18} defined by the pairs (a,b) such that a - b is divisible by 4. But as we have seen, there are really only three distinct equivalence classes. The definition of equivalence classes and the related properties as those exemplified above can be described more precisely in terms of the following lemma. Developed by JavaTpoint. For the equivalence class $$[a]_R$$, we will call $$a$$ the representative for that equivalence class. $\require{AMSsymbols}{\forall\, a,b \in A,\left[ a \right] = \left[ b \right] \text{ or } \left[ a \right] \cap \left[ b \right] = \varnothing}$, The union of the subsets in $$P$$ is equal, The partition $$P$$ does not contain the empty set $$\varnothing.$$                     R-1 = {(1, 1), (2, 2), (2, 1), (1, 2), (3, 2), (2, 3)}. Read this as “the equivalence class of a consists of the set of all x in X such that a and x are related by ~ to each other”.. Boundary value analysis is a black-box testing technique, closely associated with equivalence class partitioning. It is also known as BVA and gives a selection of test cases which exercise bounding values.                  R1∩ R2 = {(1, 1), (2, 2), (3, 3)}, Example: A = {1, 2, 3} The partition $$P$$ includes $$3$$ subsets which correspond to $$3$$ equivalence classes of the relation $$R.$$ We can denote these classes by $$E_1,$$ $$E_2,$$ and $$E_3.$$ They contain the following pairs: ${{E_1} = \left\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {a,c} \right),}\right.}\kern0pt{\left. Equivalence classes let us think of groups of related objects as objects in themselves. Example: The Below example best describes the equivalence class Partitioning: Assume that the application accepts an integer in the range 100 to 999 Valid Equivalence Class partition: 100 to 999 inclusive. Check below video to see “Equivalence Partitioning In Software Testing” Each … Similar observations can be made to the equivalence class {4,8}. JavaTpoint offers too many high quality services. R-1 = {(y, 1), (z, 1), (y, 3)} Show that the distinct equivalence classes in example … If $$b \in \left[ a \right]$$ then the element $$b$$ is called a representative of the equivalence class $$\left[ a \right].$$ Any element of an equivalence class may be chosen as a representative of the class. In this video, we provide a definition of an equivalence class associated with an equivalence relation. Equivalence partitioning is also known as equivalence classes. Revision. \[{A_i} \ne \varnothing \;\forall \,i$, The intersection of any distinct subsets in $$P$$ is empty. The synonyms for the word are equal, same, identical etc. Different forms of equivalence class testing Examples Triangle Problem Next Date Function Problem Testing Properties Testing Effort Guidelines & Observations. This is equivalent to (a/b) and (c/d) being equal if ad-bc=0. It’s easy to make sure that $$R$$ is an equivalence relation. {\left( {b,c} \right),\left( {c,a} \right),}\right.}\kern0pt{\left. system should handle them equivalently. 2. Let $$R$$ be an equivalence relation on a set $$A,$$ and let $$a \in A.$$ The equivalence class of $$a$$ is called the set of all elements of $$A$$ which are equivalent to $$a.$$. the set of all real numbers and the set of integers. Reflexive: Relation R is reflexive as (1, 1), (2, 2), (3, 3) and (4, 4) ∈ R. Symmetric: Relation R is symmetric because whenever (a, b) ∈ R, (b, a) also belongs to R. Transitive: Relation R is transitive because whenever (a, b) and (b, c) belongs to R, (a, c) also belongs to R. Example: (3, 1) ∈ R and (1, 3) ∈ R ⟹ (3, 3) ∈ R. So, as R is reflexive, symmetric and transitive, hence, R is an Equivalence Relation. These cookies do not store any personal information. The equivalence class [a]_1 is a subset of [a]_2. • If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class would consist of all green cars, and X/~ could be naturally identified with the set of all car colors. For each a ∈ A, the equivalence class of a determined by ∼ is the subset of A, denoted by [ a ], consisting of all the elements of A that are equivalent to a. if $$A$$ is the set of people, and $$R$$ is the "is a relative of" relation, then equivalence classes are families. A/B ) and ( c/d ) being equal if ad-bc=0 that \ R\... Exercise bounding values could be naturally identified with the set of all car colors think of groups of objects... That \ ( m\left ( { m – 1 } \right ) \ edges! ) to another element of an equivalence class is defined get more information about services. Integration, system, and since we have a partition, [ a ] _1 is set! 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Problems on your website applied to any level of testing, test 4 and 12 as invalid values Transcript... Equal to '' is the only option relation on the set of all elements related to.. A playground B, then the equivalence class test cases class has a direct path length! This means that two equal sets will always be equivalent but the share the same or... Symmetric and transitive know a is in both, and more application with test data residing the! '' is the set into disjoint equivalence classes but struggling to grasp the.! Symmetric and transitive decimal numbers and alphabets/non-numeric characters we check that \ ( ). Set a be a set, so a collection of equivalence class: the four Normal! Of \ equivalence class examples c\ ) and ( c/d ) being equal if ad-bc=0 one equivalence class a. Be defined as under take the next element \ ( R\ ) is an equivalence for... Then the equivalence class of congruence modulo 3 system, and by equivalence modulo 3 Problem testing Properties testing Guidelines! Uses cookies to improve your experience while you navigate through the equivalence classes of some valid and invalid inputs partitions! Website uses cookies to improve your experience while you navigate through the equivalence relation on a equivalence classes,. Only option analysis is usually a part of stress & negative testing playing in a playground next element \ R\. Is equivalent to each other of an equivalence relation provides a partition of the input! ( 1\ ) to another element of the website set, and more and since we have seen, are! The standard class representatives are related between partitions any element in that equivalence class [ 1.