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


{\displaystyle [a]}
, is the set





{\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.

View More On Wikipedia.org
  1. PhysicsRock

    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. V

    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. C

    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. L

    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. N

    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. K

    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. N

    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. K

    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. H

    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. evinda

    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. skate_nerd

    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. J

    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. I

    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. D

    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. R

    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. F

    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. F

    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. W

    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. A

    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. R

    Proving Equivalence Classes in Modular Arithmetic

    Homework Statement Suppose [d], [b] \in Z sub n.
  21. A

    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. M

    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. D

    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. M

    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. JasonRox

    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...