# What is Equivalence class: Definition and 25 Discussions

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent.
Formally, given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S, denoted by

[
a
]

{\displaystyle [a]}
, is the set

{
x

S

x

a
}

{\displaystyle \{x\in S\mid x\sim a\}}
of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by ~, and is denoted by S / ~.
When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

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1. ### Binary operation on equivalence classes

So, my approach and solution are as follows: $$[x * y] = \{ z \in M : z \sim (x * y) \}$$ Since we know that ##a * b \sim a^{\prime} * b^{\prime}## we can rewrite ##z## as ## x^{\prime} * y^{\prime} ##. Plugging this in yields:  [x * y] = \{ x^{\prime}, y^{\prime} \in M : x^{\prime} *...
2. ### A Equivalence Relation to define the tensor product of Hilbert spaces

I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces: ##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}## The definition for the equivalence relation is given in the lecture vidoe as ##(\sum_{j=1}^{J}c_j\psi_j...
3. ### MHB Equivalence Classes: Solving Cbarker1's Problem

Dear Everyone, $\newcommand{\R}{\mathbb{R}}$ I am struck in writing the equivalence classes. And the problem is this: Let ${\R}^{2}= \R \times \R$. Consider the relation $\sim$ on ${\R}^{2}$ that is given by $({x}_{1},{y}_{1}) \sim ({x}_{2},{y}_{2})$ whenever...
4. ### A Structure preserved by strong equivalence of metrics?

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...
5. ### Proving Normality of [0] in Z/3Z Quotient Group

Homework Statement I am looking at the quotient group G = Z/3Z which is additive and abelian. The equivalence classes are: [0] = {...,0,3,6,...} [1] = {...,1,4,7,...} [2] = {...,2,5,8,...} I want to prove [0] is a normal subgroup, N, by showing gng-1 = n' ∈ N for g ∈ G and n ∈ N. Since G...
6. ### B What Does the Notation = in Equivalence Classes Conclude to?

Please refer to the video at 37:02 from the link above. I am struggling with the notation "=" of the property (a) which concludes to [a]=[m]. shouldn't it be [a]⊆[m], just like [m]⊆M.
7. ### Elements of an Equivalence Class

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > Question: Let ~ be the equivalence relation on the set ℤ of integers defined by a~b if a-b is divisible by 5. Let k ∈ Em belong to the equivalence class of m, and l ∈ En belong to the equivalence...
8. ### Proof involving partitions and equivalence class

Homework Statement Show that every partition of X naturally determines an equivalence relation whose equivalence classes match the subsets from the partition. Homework Equations ( 1 ) we know that equivalence sets on X can either be disjoint or equal The Attempt at a Solution Let Ai be a...
9. ### Are all physical quantities an equivalence relation?

Consider this self-evident proposition: "If object A has the same mass as object B and object C separately, then object B has the same mass as object C." Why isn't this stated as a law, but the zeroth law of thermodynamics is? Is there a physical quantity u such that the u of A is equal to the...
10. ### MHB Each equivalence class is a power of [g]

Hello! :) I have to find an equivalence class $[g] \in \mathbb{Z_{15}}^{*}$ so that each equivalence class $\in \mathbb{Z}^{*}_{15}$ is a power of $[g]$. $\mathbb{Z}^{*}_{15}=\{[1],[2],[4],[7],[8],[11],[13],[14]\}$ I tried several powers of the above classes,and I think that there is no...
11. ### MHB Describing an equivalence class?

I am given that the relation ~ is defined on the set of real numbers by $$x$$~$$y$$ iff $$x^2=y^2$$. First part of the problem said to prove ~ is an equivalence relation, that wasn't bad. The second part asks to "Describe the equivalence classes". This just seems really vague to me. Is this a...
12. ### Equivalence class of 0 for the relation a ~ b iff 2a+3b is divisible by 5

Homework Statement ~ is a equivalence relation on integers defined as: a~b if and only if 2a+3b is divisible by 5 What is the equivalence class of 0 Homework Equations The Attempt at a Solution [0] = {0, 5n} n is an integer My reasoning for choosing 0 is that if a=0...
13. ### Quotient set of equivalence class in de Rham cohomology

Hi all, So the equivalence class X/\sim is the set of all equivalences classes [x]. I was wondering if there was a way of writing it in terms of the usual quotient operation: G/N=\{gN\ |\ g\in G\}? From what I've read, it would be something like X/\sim = X/[e]. But, since I'm looking at the de...
14. ### Mobius transformation proving equivalence class

Homework Statement I have to show that if there is a mobius transformation p such that m=p°n°p^{-1} forms an equivalence class. Homework Equations need to show that aRa, if aRb then bRa, and if aRb and bRc then aRc The Attempt at a Solution well.. for aRa I somehow need to show...
15. ### Understanding Equivalence Classes: Even and Odd Numbers in Relation to 0 and 1

Why in equivalence class of N of even number and odd number, the even number are taken as related to 0 and odd number are related as 1 i.e [0] and [1]. Instead of [0], even number can also be related to [2] or [4]? Or [2] or [4] could also be taken, as it is just an convention. Thanks.
16. ### Equivalence relation and equivalence class

i have two relations given to me which are both defined on the integers Z by relation 1: x~y if 3x^2 -y^2 is divisibale by 2 and relation 2: x~y if 3x^2 -y^2 ≥0 I have used three properties to figure out that relation 1 is eqivalence relation as it stands for all three properties i.e...
17. ### What is the Induced Equivalence Class for (2, 3) in Relation Υ?

(x1, y1)Υ(x2, y2) ⇔ x1 × y2 = x2 × y1 for all x1, x2 ∈ Z and y1, y2 ∈ Z+ have been shown to be an equivalence relation in tutorial. Specify the equivalence class [(2; 3)] as induced by Υ. i don't understand what it means by 'Specify the equivalence class [(2; 3)] as induced by Υ.'...
18. ### Proving Equivalence Class Intersection and Equality

Homework Statement I'm trying to prove that "if R is an equivalence relation on a set A, prove that if s and t are elements of A then either [s] intersect [t] = empty set, or, [s]=[t]" Homework Equations The Attempt at a Solution I know that if you were to start trying to solve...
19. ### Understanding Equivalence Classes in Integer Sets

Homework Statement Definition: If A is a set and if ~ is an equivalence relation on A, then the equivalence class of a\inA is the set {x\inA l a~x}. We write it as cl(a)Let S be the set of all integer. Given a,b \in S, define a~b if a-b is an even integer. so, the equivalent class of a...
20. ### Proving Equivalence Classes in Modular Arithmetic

Homework Statement Suppose [d], [b] \in Z sub n.
21. ### What is the Equivalence Class for the given Equivalence Relation?

Homework Statement Find the equivalence class [2] for the following equivalence relations: a) R: Z <-> Z, where xRy, iff |x| = |y| b) T: N <-> N, where xTy, iff xmod4 = ymod4 N means natural numbers etc...there wasnt the correct symbols in the latex reference Homework Equations ?? The...
22. ### Proving Equivalence of Operations on Equivalence Classes

Homework Statement Prove that if (a1, b1) ~ (a2, b2) and (c1, d1) ~ (c2, d2), then (a1, b1) + (c1, d1) ~ (a2, b2) + (c2, d2) and (a1, b1) \bullet (c1, d1) ~ (a2, b2)\bullet (c2, d2). Let [a, b] denote the equivalence class with respect to ~ of (a, b) in Z x (Z-{0}), and define Q to be the...
23. ### What the equivalence class suppose to be

I read the textbook 5 times now and I can't seem to figure out what the equivalence class suppose to be and how to find it, and i don't understand quotient set either (more importantly how to find it). I'm not familiar with any Equivalences at all if anyone can help me with it that would be...
24. ### Proving K is Not EC: Equivalence Classes

QUESTION: Let L be the language with = and one binary relation symbol E. Let epsilon be the class of all L-structures A such that E interpreted by A is an equivalence relation on |A|. K = { A in epsilon | E interpreted by A has infinitley many equivalence classes }. DEFINITIONS: EC...
25. ### Trouble understanding the path-homotopy equivalence class

Ok, I'm having trouble understanding the path-homotopy equivalence class. It's kind of blurry when they apply the operation... [f]*[g] = [f*g] ...where [f] is the path-homotopy equivalence class of f. I can see that an element in [f]*[g] is in [f*g], but not the other way around. For...