# Tagged Questions

For reflexive, symmetric and transitive relations. Use it with the tag (relations).

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### Proving Equivalence Relation $xPy$ iff $y = x + n\pi$ on The Reals

I am trying to prove that the relation P on $\mathbb{R}$ by the rule $\forall x, y \in \mathbb{R}, xPy \text{ if and only if } \exists n \in \mathbb{Z} \text{ such that }y = x+ n\pi$ From what I can ...
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### Arrow kernel in category theory and generalized equivalence relation

let $F : \mathcal{C} \rightarrow \mathcal{D}$ be a functor from two categories. It looks like that there are various notions of kernels one could define for a functor. One could define the arrow ...
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### Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$\vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m$$ The dimension of ...
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### Equivalence of Two Different Irrational Numbers Converse

Equivalence of Two Different Irrational Numbers I came across this and follow the proof, but I'm wondering how you would prove the converse, that is, if 2 irrational numbers are equivalent then their ...
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### Word for equivalence preserving transformations of equations

I am searching for a mathematical term describing an algebraic manipulation of an equation which preserves equivalence. So while adding $2$ to both sides of an equation results in an equivalent ...
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### Technical Language Usage: Verb for an Equivalence Relationship “Forgetting” an Attribute that is “Modded Away”

This question is one of English usage, but I'm sure only mathematicians can answer it for me. I want to say in a technical report that an attribute is "ignored" or "forgotten" by an equivalence ...
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### How many equivalence classes in a set of well-orders of a set

The question: Let $S$ be a set and $\operatorname{wo}(S)=\{X: X\subseteq S \land (X,\le) \text{ is a well order}\}$. Furthermore, partition $\operatorname{wo}(S)$ into equvialence classes based on ...
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### Axioms of equivalence relation in terms of the subset $R$

..An equivalence relation on $S$ is determined by the subset $R$ of $S \times S$ consisting of those pairs $\left(a,b\right)$ such that $a \sim b$. Write the axioms for an equivalence relation in ...
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### Properties of relations on Z

$$R_3 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|\frac{x-y}{5} \in \mathbb{Z}\}$$ $$R_4 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|x > y \}$$ I need to know which one is reflexive, symmetric, ...
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### A partial order with more properties than would be expected

Consider the relation: $$\langle x_1,x_2\rangle\prec\langle y_1,y_2\rangle\iff x_1,x_2,y_1,y_2\in\Bbb N\wedge x_1y_2<x_2y_1.$$ This is usually used for defining the (positive) fractions $\Bbb Q^+$...
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### Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single maximal ...
I would like to know how to calculate this integral n check its nature : $$A= \int_1^\infty \frac{e^{-t}}{1+t^{2}} dt .$$ I think I figured it out , we can do it with Equivalence like this (since ...
### On $N \times N$ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$. Show that $R$ is an equivalence relation.
On $N \times N$ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$ a. Show that $R$ is an equivalence relation. My attempt: By definition 6.2.3 $R$ is an equivalence ...