Articles of equivalence relations

Measure spaces s.t. $\mathcal{L}^1 = L^1$

I have two questions: 1, Give an example of a measure space such that $L^{1}(X,\mathcal{A},\mu) = \mathcal{L}^{1}(X,\mathcal{A},\mu)$. 2, State, and prove, a condition on $\mu$ which is equivalent to the identity in 1. Attempt at solution: 1, Let $X = \mathbb{N}$, $\mathcal{A} = \mathcal{P}(\mathbb{N})$. The idea is that every $n\in \mathbb{N}$ defines a function. But […]

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked – I searched but couldn’t find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) $B$ that is block diagonal form: I know both of these. I’m looking for a similarity matrix $Q$ such that $B = […]

Equivalence relation on a proper class

We define cardinality as an equivalence relation on sets. But the class of all sets is not a set, so how do we do that? In particular, I’m interested in the proposition that equivalence classes form a partition of the initial set. It seems like it can be translated to cardinality, but I do not […]

Is there a name for this property: If $a\sim b$ and $c\sim b$ then $a\sim c$?

Let $X$ be a set with a binary relation $\sim$, such that for all $a$, $b$, and $c$ in $X$: If $a\sim b$ and $c\sim b$ then $a\sim c$ Is anyone familiar with this property of a binary relation? Does it have a name? Does it have any interesting properties?

Why not just define equivalence relations on objects and morphisms for equivalent categories?

My question is regarding this blog post https://unapologetic.wordpress.com/2007/05/30/equivalence-of-categories/ which I will paraphrase below: The author gives the example of a category $\mathscr{C}$ with just one object and the identity morphism, and another category $\mathscr{D}$ consisting of two objects and 4 morphisms (2 identity and 2 non-identity). After some discussion the author reveals that all 4 […]

connected components equivalence relation

Prove that the relation $x \sim y$ iff $y$ is an element of the connected component of $x$ is an equivalence relation. This question is confusing me, do I simply go about showing the relation is reflexive, symmetric, and transitive? I don’t really see how to do this for this question. Any suggestions or hints […]

For a set of symmetric matrices $A_i$ of order p, show that if the sum of their ranks is p, $A_iA_j=0$

Here’s what I know. Matrices $A_i$ for $i=1,…,k$ are all symmetric p by p matrices. $\sum\limits_{i=1}^k A_i = I_p$ where $I_p$ is the p by p identity matrix $\sum\limits_{i=1}^k rank(A_i) = p$ With this, I have to find a way to show that for all $i \neq j$, $A_iA_j=0$. I assume this is solved by […]

Is there a name for relations with this property?

Is there a name for relations $\rho : X \rightarrow Y$ such that for all $x,x’ \in X$ and all $y,y’ \in Y$ we have that the following conditions $$xy \in \rho$$ $$x’y \in \rho$$ $$xy’ \in \rho$$ imply that $$x’y’ \in \rho?$$ Here’s a couple of alternative characterizations. Define $\rho(x) = \{y\in Y \,|\, […]

When is the topological closure of an equivalence relation automatically an equivalence relation?

Let $X$ be a topological space, not necessarily nice, let $R$ be a subspace of $X \times X$, and let $\overline{R}$ be the closure of $R$ in $X \times X$. Then: If $R$ is a reflexive binary relation on $X$, then $\overline{R}$ is also a reflexive binary relation on $X$. If $R$ is a symmetric […]

How to prove that equality is an equivalence relation?

Probably, it’s a elementary question, but I would like some explanation. Everyone knows that equality relation is (i) reflexive, (ii) symmetric and (iii) transitive, that is, satisfies (i) $x=x$; (ii) $x=y\Rightarrow y=x$; (iii) $x=y$ and $y=z\Rightarrow x=z$. I’m interested in to deduce these properties from some appropriate definition of equality. Fraleigh’s algebra book presents the […]