Why are equivalence classes useful

Similar examples can be found in any field of mathematics. I was originally going to use building the tangent space in differential geometry, but switched to surfaces as being more accessible.

Equivalence relations are one of the most powerful tools in the mathematician's tool box. You will find them used everywhere. One instance in math where the notion of an equivalence relation is particularly useful, and makes the axioms of an equivalence relation seem natural, is considering a set modulo some equivalence relation.

It is also useful when you have some sets with some type of structure groups, rings, topological spaces and you want some way to think of them as being the same.

We usually do this by defining an appropriate 'structure preserving function' between them and intuitively these functions should satisfy the axioms of an equivalence relation. In short an equivalence relation is a nice way to make precise the manner in which we can regard two things as the same even if they are not "equal". For the laymen example suppose I am talking about all the shoes I own, but I don't want to talk about distinct shoes, I just want to talk about all the pairs of shoes I own.

My blue nike that goes on my left foot is not actually the same shoe as my blue nike that goes on my right foot, but if I consider my shoes modulo the equivalence relation that they belong to the same pair - both of my blue nikes just became the same shoe.

In essence I can talk about, or prove things about, both shoes by just considering one of them. Because modulo the equivalence relation, they are "equal". Edit: added after original post. Also, something I noticed as I was advancing into higher more abstract mathematics which I am still doing! We love to take some normal concept such as 'equality' or 'distance' and say 'what are the properties that are inherent to distance' or 'what are the properties that are inherent to the notion of equality'.

If you sit down and brainstorm for each, you will likely come up with the axioms for a metric space in the distance case, and the axioms of an equivalence relation in the equality case. Then, mathematicians love to say 'what happens in the more general setting where more general objects obey these properties? In this sense, the "motivation" would be more along the lines of curiosity. Then, if the theory is turning out to be fruitful and producing interesting results, that further motivates the study of this new 'generalized' notion.

I actually haven't studied graph theory, so I can speak to much about your example regarding connected components, but my hunch is that using the equivalence relation is 'just general enough' in the sense that, by avoiding using an equivalence relation you could, at some point, be limiting your self by not being general enough or get wacky results because things are too general. Me personally, everything I study is algebraic - and equivalence relations are imperative.

They help to study smaller algebraic structures without forgetting the old structure it came from - and again help make precise what it means for two algebraic structures to be equal. Equivalence relations are a way to describe objects that are effectively the same in a given context.

Then there will be six connected components, each corresponding to an equivalence class. If you zoom out far enough, then each of these components looks like a single object, and it is more interesting to look at how these objects relate to each other than to look at the infinitely many but effectively the same possible details. An equivalence relation is a very basic kind of structure that occurs all through different areas of mathematics.

You've mentioned two examples: modular arithmetic and connected components of graphs. There are many others. Yes, it would be possible to develop each one by itself without using the term "equivalence relation". But mathematicians are supremely lazy: they don't like to duplicate effort. And once you have the concept of equivalence relation, all you have to do is recognize that something is an equivalence relation and a whole set of tools is automatically available without further effort.

Although pretty much everything that needs to be said has already been addressed much of it in MJD's excellent answer , I thought I'd share an excerpt from Princeton's Companion to Mathematics that was very helpful to me when I first encountered equivalence relations, equivalence classes, and all that jazz. There are many situations in mathematics where one wishes to regard different objects as "essentially the same," and to help us make this idea precise there is a very important class of relations known as equivalence relations.

Here are two examples. First, in elementary geometry one sometimes cares about shapes but not about sizes. Two shapes are said to be similar if one can be transformed into the other by a combination of reflections, rotations, translations, and enlargements.

See the similar shapes in the figure below. The relation "is similar to" is an equivalence relation. What exactly is it that these two relations have in common?

The answer is that they both take a set in the first case the set of all geometrical shapes, and in the second the set of all whole numbers and split it into parts, called equivalence classes , where each part consists of objects that one wishes to regard as essentially the same. One of the main uses of equivalence relations is to make precise the notion of "quotient" constructions [as alluded to in MJD's answer]. There are a lot of great answers in here, but I thought I would include an important equivalence relation that arises in linear algebra.

When two matrices are similar, they represent the same linear transformation though possibly using different bases. So why might we care about this?

Well it can make computations easy for one. If you are presented with a matrix, you can determine what the underlying linear transformation is and there is often an "optimal" basis to represent that linear transformation that makes the corresponding matrix as simple as possible i.

You can then use that particular matrix to do all of the computations you have in mind and save your self some time. Equivalence relations are important because of the fundamental theorem of equivalence relations which shows every equivalence relation is a partition of the set and vice versa.

Because partitioning is a very common idea in both pure and applied mathematics equivalence relations occur naturally but have the added benefit of allowing you to exploit the transitive, reflexive and symmetric properties of the relation. I would also point out that "standard graph theory language" as you called it is to use equivalence relations.

Very few theorems in mathematics are given the title "fundamental" and they show up in algebraic settings, logic, and even cutting and pasting surfaces together. It's very powerful and widely applicable modelling tool which is why they're ubiquitous in higher mathematics. By following a set of guidelines while implementing the process of testing , the team of testers can ensure better outputs from the tests and make sure all scenarios are being tested accurately.

Boundary value analysis and equivalence class testing are two strategies used for test case designing in black box testing, which makes it crucial for us to differentiate them from one another and define their specific relevance in software testing. The differences between these two are:. A type of specification based testing or black box testing technique , Equivalence Partitioning or Equivalence Class Testing is a widely used method that decreases the number of possible test cases that are required to a software product.

Moreover, its ability to generate greater testing coverage, without compromising time and efforts, makes Equivalence Class Testing a popular testing technique worldwide. We deliver.

Get the best of Professional QA in your inbox. Toggle navigation. The Definition of an Equivalence Class We have indicated that an equivalence relation on a set is a relation with a certain combination of properties reflexive, symmetric, and transitive that allow us to sort the elements of the set into certain classes. Progress Check 7. Answer Add texts here. Do not delete this text first. Determine the equivalence class of 0. Corollary 7. Exercise 7. Determine all of the congruence classes for the relation of congruence modulo 5 on the set of integers.

In Progress Check 7. Also, see Exercise 9 in Section 7. See Exercise 13 in Section 7. Determine all the distinct equivalence classes for this equivalence relation. In Exercise 15 of Section 7. What we did for the specific partition in Part 12b can be done for any partition of a set. Equivalence Relations on a Set of Matrices. The following exercises require a knowledge of elementary linear algebra. Justify your conclusion. De Morgan's Laws 4. Mixed Quantifiers 5. Logic and Sets 6.

Families of Sets 2 Proofs 1. Direct Proofs 2. Divisibility 3. Existence proofs 4. Induction 5. Uniqueness Arguments 6. Indirect Proof 3 Number Theory 1.