# Tagged Questions

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### Is $^\mathbb{N}\mathbb{R}$ $\sim$ $^\mathbb{R}\mathbb{N}$? [duplicate]

Is $^\mathbb{N}\mathbb{R}$ $\sim$ $^\mathbb{R}\mathbb{N}$? I know you have to use Cantor-Bernstein, and prove both directions, but i don't know how to start the proof
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### Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R?

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R? I don't even know how to start. I know to be a equivalence relation R must be reflexive, symmetric and ...
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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. ...
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 ...