# Tagged Questions

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### Doubt pertaining to this Equivalence Relation.

$1$. True or false? If $R$ is an equivalence relation on a set $S$ and it has only finitely many equivalence classes altogether, then $S$ itself is a finite set. I think the answer is true ...
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### Equivalence relation- Equivalence Classes And Partitions

I had the following question A is a finite set and $R \subseteq A \times A$ is a equivalence relation. Prove that $|A|$ is odd iff $|R|$ is odd. I am trying to find a general formula for this ...
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### equivalence classes and cardinality

I need to prove that every equivalence class created by the equivalnce relation $\sim$ on $\mathbb{R}$, that is defined by: $a\sim b \Leftrightarrow (a-b) \in \mathbb{Q}$, is $\aleph_0$. Furthermore, ...
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### The null set as the underlying set of a relation structure

Can the null-set underly a relation structure? Can it underly a well-order? If so, would such a well-order have order-type $0$? (Can one even have an order-type of $0$?) My motivation for this ...
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### Equivalence and Order Relations

I have the following problem: Provide an example of a set $S$ and a relation $\sim$ on $S$ such that $\sim$ is both an equivalence relation and an order relation. Conjecture for which sets and this ...
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### Sets Modulo Equivalence Relations

I am stuck on this question and would greatly appreciate any help: Recall, for an arbitrary set $S$ and equivalence relation $\equiv$ on $S$, $S/\equiv$ denotes the set of equivalence classes in $S$. ...
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### Colored Picture for Equivalence Classes, Relations, Partitions, ..

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
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### What does an equivalence class look like?

Let $\mathbb{R}^2 = \{Q = (a,b) | a,b\in \mathbb{R}\}$. Prove that if $q_1 = (a_1,b_1)$ and $q_2=(a_2,b_2)$ are equivalent, meaning $a_1^2+b_1^2 = a_2^2 +b_2^2$, then this gives an equivalence ...
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I would appreciate any help available for the following problem: Let $S$ be a set. Let $T$ be the set of all relations on $S$. Construct a relation $\equiv$ on $T$ in the following way: for $\sim, ... 3answers 175 views ### Can we take images of equivalence relations? Given a function$f : X \rightarrow Y$, it is well-known that we can take the image under$f$of any subset$A \subseteq X$, and we can take the preimage under$f$of any subset$A \subseteq Y$. This ... 1answer 104 views ### Equivalence relation question with cardinality and countability$A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $Let$A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $What is the cardinality of$[\pi]_S$? Prove that the quotient group$\mathbb R/S$is uncountable. Well I think that cardinality is ... 1answer 124 views ### Prove or disprove question with equivalence relation, classes and quotient group Let$A$be a set and$R$an equivalence relation on$A$. Prove or disprove: If$A$is countable then all the equivalence classes of$R$are countable. If$A$isn't countable then the ... 3answers 274 views ### Equivalence relation question with functions We'll define on the set:$A=\Bbb R^{[0,1]}$the relation$R$by$fRg$if$f(0)=g(0)$. Make sure it's an equivilence relation. What is$[\cos x]$? Describe all the equivalence classes ... 3answers 63 views ### Proof of equivalence relation on a set Let$X = \{(a,b) \mid a,b \in \Bbb Z; b \ne 0\}$. We define$(a,b)\mathrel R (c,d)$iff$ad = bc$. Prove that$R$is an equivalence relation on the set$X$. Which known set do the equivalence classes ... 2answers 75 views ### Properties of the relation$R=A\times B \cup B\times A$A is a set. Let$B\subsetneq A$.$R=A\times B \cup B\times A$Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, ... 1answer 62 views ### Properties of the relation$R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}A= \Bbb R \\ R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, (g)equivalence ... 1answer 65 views ### Equivalence relation$g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$Let$(G, \cdot)$be a group with an Identity element$e$. (i) A relation on$G$is defined through$g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$. Show that$\sim$is a equivalence relation and ... 1answer 32 views ### Formulate a relation$R$between$2$sets$A$and$B$Let$A$and$B$be$2$sets of real numbers. How can I formulate the following entence, in mathematical terms, not plain english. IF At least one Element$x$of$A$is equal to one element$y$of ... 0answers 59 views ### What is this equivalence relation explicitly? Let$S \colon = \{ \ (x,y) \in \mathbf{R}^2 \ | \ \ y = x +1, \ \ 0 < x < 1 \ \}$, and let$T$be the intersection of all the equivalence relation on the plane that contain$S$. Then how ... 1answer 128 views ### equivalence relation composition problem Let$R_1$,$R_2$be two equivalence relations on$X$, prove that$R_1\circ R_2$is an equivalence relation if and only if$R_1\circ R_2= R_2\circ R_1$First I´m trying to prove that$R_1\circ R_2= ...
Let $R_1$, $R_2$ be 2 equivalence relations on $X$; prove that $R_1\cup R_2$ is an equivalence relation on $X$ if and only if $R_1\cup R_2=R_1\circ R_2$ I really don´t have any idea how to do it, I ...
Let $R$ and $S$ be equivalence relations on X so that $X/R$=$X/S$, prove that $R=S$ how can I solve this problem?