1
vote
2answers
36 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
0
votes
1answer
33 views

Is a relation$ R=\{(a,a):a\in I\}$ said to be reflexive, symmetric and transitive?

Consider a relation $R=\{(a,a): a\in A\}$ where $A=\{1,2,3\}$. i.e $R=\{(1,1),(2,2),(3,3)\}$ This clearly reflexive but is it necessary that all such relations are necessary to be symmetric and ...
2
votes
1answer
30 views

Generated equivalence relations in logics

Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. ...
0
votes
3answers
286 views

Quotient spaces in linear algebra

There's a statement in some notes I'm reading that goes like this: "...$V/U$ is a 'simplified version' of $V$ where the elements of $U$ are ignored" ($V$ and $U$ are vector spaces). I'm still ...
1
vote
2answers
60 views

Equivalence relation proofs: general or specific?

I'm confused about whether a specific example must exist to prove an aspect of an equivalence relation. For example: if a set, $A$, only contains one element, $A = \{1\}$, and a relation, $R$, on ...
2
votes
1answer
113 views

Why is the term “composition” used to mean a certain binary operation on the set of relations on a given set?

I looked for material relating to compositions of equivalence relations, and was surprised to find the claim (here) that the composition of equivalence relations is not necessarily again an ...