# Tagged Questions

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

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### Why isn't reflexivity redundant in the definition of equivalence relation?

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity. Doesn't symmetry and transitivity implies reflexivity? Consider the following argument. For any $a$ ...
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### When are algebraic expressions equivalent?

This question arose when I was going to determine the domain for $f \circ f(x)$. Let $f(x) = \dfrac{1-x}{1+x}$. $f \circ f(x) = x, \quad$ But the domain is not $\mathbb{R}$ because $f(x)$ is undefined ...
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### What do you get with this equivalence relationship for all $\mathbb{Q}$ sequences

Consider all $\mathbb{Q}$ Cauchy sequences with this equivalence relationship $\{x_n\} \sim \{y_n\} \iff \{x_n-y_n\} \rightarrow 0$ Then you get all real numbers as an equivalence class with this ...
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### Understanding equivalence class, equivalence relation, partition

Im having difficulty grasping a couple of set theory concepts, specifically concepts dealing with relations. Here are the ones I'm having trouble with and their definitions. 1) The collection of ...
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### Dependence of Axioms of Equivalence Relation?

This question is problem 11(a) in chapter 1 in 'Topics in Algebra' by I.N. Herstein. These are the properties of equivalence relation given in this book. Prop 1 $a \sim a$ Prop 2 $a \sim b$ ...
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### Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
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### why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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### Is the quotient map a homotopy equivalence?

It is well known that, if $A \subset X$ is a reasonable contractible subspace, then the quotient map $X \to X/A$ is a homotopy equivalence ("reasonable" means that the pair $(X,A)$ has the homotopy ...
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### 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 ...
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### Are these two equivalence relations the same ? (Equivalence relation defined using subgroups of a group)

I was solving the book Abstract Algebra by Charles Pinter. In one of the questions I was stuck. The question is as follows :- $H$ is a subgroup of group $G$.Two equivalence relations on $G$ are ...
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### Definition of “quotient set”

I searched and searched about quotient set and cannot figure out what it is. At the beginning, I thought it was the same as partitions, but now I'm confused. Can someone show some examples and explain?...
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### Quotient of a graph?

I want to understand quotient of a graph (also called quotient graph), my teacher says that the terms quotient of a graph and a modulo of a graph should be synonyms (even though modulo of a graph ...
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### How to simply show that there are “78 'strict ordinal' 2x2 game matrices”

In "Theory of Moves", Steven J. Brams analyses two-player games with two strategies per player, where each player can totally rank his payoffs, although payoffs need not be comparable among players. ...
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### Solve for ? - undetermined inequality symbol

So I was solving a problem in Rudin (chapter 3 #16, to be specific) and I realized how convenient it would be to have a symbol that represented an undetermined equivalence relationship. As an example ...
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### Number of equivalence relations on a finite set

I need a hint for computing the number of ways in which all the equivalent classes on a set of $n$ elements can be realized. For example, if the set has 2 elements ${a,b}$, then there are 2 possible ...
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### Prove that $(H,\circ)$ is a subgroup of the group $(G, \circ)$

Question: Let $(G, \circ)$ be a group and $H$ be a non-empty subset of $G$. A relation $\rho$ defined on $G$ by $$a\,\rho\ b\quad \text{if and only if}\quad a\circ b^{-1}\in H$$ for $a,b\in G$, is an ...
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### The fibres of a map form a partition of the domain.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
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### What's the name for the equivalence induced by a function on its domain?

Any function $f$ with domain $X$ induces an equivalence relation on $X$, with classes $$\{f^{-1}(\{y\})\,:\, y \in \operatorname{im}f\;\} .$$ Is there a name for this equivalence? Thanks!
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### Equivalence relations on classes instead of sets

Can someone please explain to me how to deal with equivalence relations on classes instead of sets? Is there some sort of generalisation of relations? Thank you
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### How do I work with a relation that is a set of 4-tuples?

Define the relation $\sim$ on $\mathbb{Z}\times\mathbb{Z}$ by $(a,b)\sim(c,d)$ if $a-c=b-d$. Show that $\sim$ is an equivalence relation. What is the equivalence class of $(1,2)$? I'm not sure how ...
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### About proving that $Aut(\mathbb{Z}_n)\simeq \mathbb{Z}_n^\times$.

I need to prove that $$Aut(\mathbb{Z}_n) \simeq \mathbb{Z}_n^\times.$$ My definition of $\mathbb{Z}_n$ is that $$\mathbb{Z}_n =\{\bar{m}: m\in \mathbb{Z}\}$$ where $\bar{m}$ is the equivalence ...
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### Why are equivalence classes called “classes” and not “sets”?

Why are equivalence classes called so and not equivalence sets? I am kind of not able to find the difference between a class and a set. What properties that a set have that a class cannot have? It ...
### Are there any distinct $a, b$ s.t. $a + x$ prime $\Longleftrightarrow$ $b + x$ prime?
Are there any distinct $a, b \in \mathbb{N}$ s.t. $a + x$ prime $\Longleftrightarrow$ $b + x$ prime for all $x \in \mathbb{N_0}$? I can show there are no coprime $a,b$ using Dirichlet's theorem: ...
### $5 \mid n^2 - m^2$ is an equivalence relation
How can I show this is an equivalence relation: $$n \operatorname{R} m \Longleftrightarrow n^2 - m^2 \textrm{ is divisible by } 5$$ I know equivalence relations are symmetric, reflexive and ...