Questions tagged [equivalence-relations]
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.
3,175 questions
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uniqueness of a map and universal property
I am trying to prove the following statement:
For any two non-empty sets $X,Y$, an equivalence relation $E$ on $X$, and an injective function $h:X/E\to Y$, show that there exists a unique function $H:...
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Are these theories complete
Let $T$ be a $\mathcal{L}$-theory defining $E$ as an equivalence relation and $T'$ be a $\mathcal{L}$-theory stating that $E$ is an equivalence relation with infinitely many classes. Are $T$ and $T'$ ...
anon
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Equivalence relations used in the proof of the Vitali Theorem
In the proof of Vitali Theorem, to show the existence of sets that are not Lebesgue measurable, the following relation is used to construct the Vitali set:
On $[0,1]$ I’ve got the relation $\sim$ ...
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Is it $\aleph_0$-categorical theory
Let $\mathcal{L}=\{E\}$ where E is a binary relation. Let $T$ be a $\mathcal{L}$-theory stating that $E$ is an equivalence relation with infinitely many classes, all of which are infinite. Show that $...
anon
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Why proving the well-definedness for the quotient map is different here?
I'm reading Aluffi's Chapter 0, here:
We consider then an equivalence relation $\sim$ on (the set
underlying) a group $G$; we seek a group $G/\sim$ and a group
homomorphism $\pi: G \to G/\sim$ ...
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cyclic permutations and equivalence class in finite sets
I already have a solution of the problem, but it was very ugly and I wanna know if someone have a more clean proof.
This is the problem: let $A$ be a finite set and $\sim$ an equivalence relation such ...
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If $f:X→Y$ is injective and compatible with equivalence relations $E_X$ and $E_Y$ is $\widetilde f: X/E_X→Y/E_Y$ injective?
Let $f$ be a map from a set $X$ into a set $Y$; let $\mathcal E_X$ be an equivalence relation on $X$ and let $\mathcal E_Y$ be an equivalence relation on $Y$. We say that $f$ is compatible with $\...
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Transitive property [closed]
Can a relation with only 1 element be called a transitive relation? My module says so but i feel like it does not make any sense to define transitive property for a single element relation
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Quotients of compact metric spaces whose projection maps are local homeomorphisms
Relevant Definitions
A map of topological spaces $f: X \to Y$ is said to be a local homeomorphism if for each $x \in X$ there exists some open set $U \subseteq X$ with $x \in U$ such that $f(U) \...
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Quantifier elimination of a language with 1 binary equivalence relation
At the moment I am taking a course in logic (which includes an extensive amount of model thoery). In the last lecture we learned about quantifier elimination. On the last homework assignment we were ...
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Is it Enough for Axioms to Hold over an Equivalence Relation?
this is an odd question, but I am quite curious about this. I have only slightly more than a quarter's worth of understanding of abstract algebra, but I was wondering, when we define an object such as ...
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I want to check whether the following relation is Transitive relation or not.
Q$)$ If R is a relation defined as $$R=\{(a,b):a\leq b^{4}\}$$ where $a,b \in \mathbb{N}$ then check whether the above relation is Transitive Relation or not
My Approach
I already checked whether ...
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Ordering Input Variables and Output Variables for a Jacobian
In this video, it is shown that Jacobian Matrices of entities such as an LU Decomposition or an Eigenproblem can be computed. However, relative to, i.e., the matrix squaring function, it is less ...
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How closely related are homogeneous coordinated point equivalence classes and algebraic ideals?
This first definition is according to Weyl's The Classical Groups:
An ideal is a subset $\lbrace0\rbrace\ne\mathfrak{a}\subseteq\mathbb{r}$ of the ring $\mathbb{r}$ such that for $\alpha,\beta\in\...
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Why is the quotient category on objects not well-defined?
I am taking classes on Category theory, and my professor recently talked about weak equivalences, congruences on a category, quotient categories, category of fractions...
Every one of those ...
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