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.
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Timelike vector equivalence relation [closed]
I'm a little bit stuck with the demonstration of the equivalence relation between timelike vectors, specifically in the transitive property. These are the assumtions:
$\tau =\left \{ v\in \mathbb{R}^4:...
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Open Equivalence relation induces discrete topology on the quotient
Let $X$ be a topological space and $\sim$ an equivalence relation open as a subset of $X\times X$. I need to show that the quotient $X/\sim$ then has the discrete topology.
That it, I need to show ...
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Complete Lattices and the Injectivity of the Restriction $f|_S$ - Verification of Proof
Attempt (General Case)
Conjecture: I want to show that if $X$ and $Y$ are nonempty sets, $(X, \leq)$ is a complete lattice, and $f: X \to Y$ is any well-defined function, then there exists a nonempty ...
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Can equivalence relations have extra non-trivial properties?
The theory of equivalence relations can be axiomatized by 3 equality-free universal sentences, namely:
1.$xRx$
2.$xRy \rightarrow yRx$
3.$(xRy \land yRz) \rightarrow xRz$.
Now, certainly, we can add ...
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Does the quotient set uniquely determine arbitrary homogenous relations?
It seems totally reasonable to me to generalize the quotient of equivalence relations to general ones:
Given a binary relation $R$ over sets $X, Y$, each $x \in X$ has an $R$-class, denoted $[x]_R = \...
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Axioms of countability and quotient topology
I am trying to do the following exercise regarding axioms of countability and quotient topology:
In $\mathbb{R}^2$ (with the euclidean topology) consider the equivalence relation: $(x,y) \sim (x',y') \...
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Interpreting Row Equivalence in the Context of Maps
I have been exploring linear algebra for the last month and a half. My textbook, Linear Algebra by Jim Hefferon (2020), drives home the point that matrices are primarily a tool for representing maps. ...
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Types of relations- Reflexive, Symmetric, Transitive, Identity, Universal, Null and Equivalence relations
By the definition of it, it appears that a relation can be an identity relation or a reflexive relation only if the relation is on a single set (ie. it is subset of cartesian product of a set with ...
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Equivalence classes and sequences
Suppose I'm working in $\mathbb{R}$ and I have the equivalence relation, $a\mathcal{R} b \leftrightarrow a \text{ mod } 10=b \text{ mod } 10$. Suppose now that I'm given a sequence $\{a_n\}_n\in \...
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Equivalence Relations in the colimit of Sets
The Stacks project mentions colimit in the Sets and introduces the following equivalence relationship: $m_{i} \sim m_{i^{'}}$ if $m_{i} \in M_{i}$, $m_{i^{'}} \in M_{i^{'}}$ and $M(\varphi)(m_{i}) = ...
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Beginner in equivalence relations. Example.
I'm approaching the study of pure Algebra, starting from equivalence relations, sets and quotient sets. I have a background in Algebra, but it's only a bit of linear algebra and I never worked with ...
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Equivalent functions are both $L_1$
Let $f(x) \sim g(x)$ and $f(x) \in L_1(\mathbb{R})$. Then it is stated that $g(x) \in L_1(\mathbb{R})$, is that true?
I feel like the answer is positive, but I can not bound $g(x)$ from above with ...
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How to interpret equivalence relations graphically.
Imagine a relation $R$ as a subset of the cartesian product of the real numbers with itself. This relation can be interpreted as a graph.
If R is an equivalence relation
Reflexivity can be ...
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Axler Example 3.100 on quotient spaces
In Axler's linear algebra textbook, he gives an example of a quotient space as $\mathbb{R}^3/U$, where $U$ is a line in $\mathbb{R}^3$ containing the origin. The quotient, he claims, is the set of ...
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Is slope $m=(y-y_1)/(x-x_1)$ same as $y-y_1 = m(x-x_1)$?
My question is rather subtle. In the context of functional equivalence, that is, they both having same domain and same output. The prior is undefined at the point $(x_1,y_1), m=0/0$. But multiplying ...
