Questions tagged [equivalence-relations]
Olivia Zamora 
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For questions about relations that are reflexive, symmetric, and transitive. These are relations that model a sense of "equality" between elements of a set. Consider also using the (relation) tag.
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Showing an equivalence relation of Orbits
I want to show an equivalence relation of Orbits of a flow. Let $ y'=f(y)$ with $y \in\mathbb{R}^n $, f locally Lipschitz, $\phi$ the flow of the ODE. Show that: $y_1\sim y_2 \iff \exists x_2\in I_{... real-analysis ordinary-differential-equations analysis equivalence-relations- 1
The quotient space of the equivalence relation $x^2-y = x'^2-y'$
On the euclidean plane $\mathbb{R}^2$ we define an equivalence relation by $(x,y) \sim (x',y')$ iff $x^2 -y = x'^2 - y'$. In other words, I consider the set $$ \mathbb{R}/{\sim} := \{ [(x,y)] \mid (x,... general-topology equivalence-relations quotient-spaces- 59
"Continuous" equivalence relation generated by a continuous weak order.
Let continuous weak order $\succsim\supset\ge$ be a subset of $\mathbb R^2\times\mathbb R^2$. Let $\sim=\succsim\cap\precsim$. A binary relation is an equivalence relation iff it is transitive, ... real-analysis order-theory equivalence-relations- 513
What does the author mean here by the congruence relation generated by -?
I am reading up on Wheel Theory using the notes found at There really doesn't seem to be many online notes for this topic. It starts by motivating ... equivalence-relations congruence-relations wheel-theory- 6,260
Describing quotient set under the equivalence relation $x \sim y$ iff $\sin(x) = \sin(y)$.
I don't have any trouble showing that for any function $f: X \to Y$, the relation $x \sim y$ if and only if $f(x) = f(y)$ is an equivalence relation, all properties of which follow from reflexivity, ... solution-verification equivalence-relations- 401
Proof verification of Set theory problem
Given an infinite set $X$ consider the set of sequences $X^\mathbb{N}$ and the map $s$ that shifts all elements up by $1$ or rather $(a_i)\mapsto (a_{i+1})$. The problem asks to find a map $s_{1/2}$ ... elementary-set-theory solution-verification equivalence-relations- 1,540
Improper Equivalence of Integral Quadric Forms is not an Equivalence Relation
An integral quadric form is some instance of $f(x,y)=ax^2+bxy+cy^2$, with $a,b,c$ integers. Let $f(x,y),g(x,y)$ be two integral quadric forms. Then we say that they are improperly equivalent, denoted ... number-theory equivalence-relations quadratic-forms- 165
Is AC necessary here?
I was trying to prove this statement: Let $f:X\to Y$ a surjective map and $g: X\to Z$ such that $$\forall\,x,y\in X: f(x)=f(y)\implies g(x)=g(y).$$ Then exists a unique map $h:Y\to Z$ such that $g=h\... equivalence-relations axiom-of-choice- 351
Showing that $x^3 + y = y^3 + x$ is an equivalence relation
I am asked to prove that: $x^3 + y = y^3 + x$ is an equivalence relation. So far I have the following: Reflexive: $m^3 +m = m^3 +m$ Symmetric: $m^3 + n = n^3 + m \rightarrow n^3 + m = m^3 + n$ ... discrete-mathematics relations boolean-algebra equivalence-relations- 53
Help finding mistake in proof involving the quotient map.
Consider the plane $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ with the product topology which has basis consisting of all open squares of the form $$\tag{1} \left]a,b\right[ \times \left]c,d\right[ \... real-analysis general-topology discrete-mathematics equivalence-relations quotient-spaces- 159
Proving that the quotient of a set by an equivalent relation is a partition
I need to show that the quotient of a set $S$ with respect to the equivalence relation $\sim$ is a partition of $S$. To show this, we will denote the quotient by $P_\sim.$ Note that $$ P_\sim = \{[a]_\... elementary-set-theory solution-verification equivalence-relations set-partition- 18
Characterize open sets of this quotient topology.
Let $X$ the quotient space obtained from $\mathbb{R}\times\{0,1\}$ identifying $(x,0)\sim(x,1)$ if $|x|>1$. Which are the open sets of this quotient topology? First, I've made the next drawing to ... general-topology equivalence-relations quotient-spaces- 1,721
Objects that change their equivalence class under some transformation
Suppose there is a set of objects on which we can define an equivalence relation. Under some transformations of the space on which the objects are defined, these objects may change their equivalence ... equivalence-relations- 21
When all commutant (centralizer) subgroups are abelian
I have seen the following problem in chapter 9 of Abstract Algebra by Dan Saracino: Let $G$ be a group and for $a,b \in G$ let $a\ R\ b$ mean that $ab=ba$. Must $R$ be an equivalence relation on $G$? ... abstract-algebra group-theory abelian-groups equivalence-relations- 1,219
Intuition behind the definition of matrix similarity/equivalence? [duplicate]
Given two matrices $A$ and $B$, they are similar if: $$B=P^{-1}AP $$ Furthermore, if they are similar they are relative to the same linear transformation (equivalent). However the proof I've checked ... linear-algebra matrices linear-transformations equivalence-relations similar-matrices- 413
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