Are sets always represented by upper case letters? -- and understanding a bit about equivalance relations I'm trying to solve q question which states:
Let $\leq$ be a preorder relation on a set $X$, and $E=${$(x,y)\in X:x\leq y$ and $y\leq x$} the corresponding equivalence relation on $X$. Describe $E$ if  $X$ is the set of finite subsets of a fixed infinite set $U$, and $\leq$ is the inclusion relation (i.e $x \leq y$ iff $x\subset y$)
I naively said that $E=${$(x,y) \in X$:$x\subset y$ and $y\subset x$}$=${$(x,y) \in X:x=y$}
I have a few concerns however, the question says $x \leq y$ iff $x\subset y$, can we just assume that x and y are sets even though they are represented by lower case letters thus be able to use the $\subset$ relation on them (it was not specified that elements of $X$ are sets)?
Secondly, what is the significance of stating that X is the set of finite subsets of a fixed infinite set U?
Thanks
 A: *

*No, set can also be denoted by lower case letters. In a set theoretic fundation everything is a set. Also, in mathematics, in general, anything might be denoted by anything as long the definitions are clear to follow.

*Well, this is just a particular example which can arise in more situation, e.g. in model theory of first order logic.

A: We use mostly arbitrary symbols for sets (or other objects). Sometimes $c,C,\gamma, \Gamma,\mathcal C, \mathbf C$ occur in the same text and all denote sets, albeit at possibly different "levels". There is no intrinsic rule, but mnemotechnical considerations at work.
Here, we first talk about sets $X$ and $E$ and denote elements of $X$ that are not necessarily sets as $x,y$. So far, so good. No dditional "inner" structure of $x,y$ is needed or investigated, all we are given is  that there is a relation $\le $ on $X$ that fulfills the axioms of a preorder. So for particular $x,y$ it may be the case that $x\le y$ or not.
In an example it is of course possible that $X$ is not just a set, but a set o fsets, i.e. its elements such as $x,y$ are sets themselves. Here, for example $U$ should be infinite, so maybe $U=\mathbb N$, and $X$ the set of finite subsets of $U$, so the elements of $X$ are the sets $\emptyset, \{1\}, \{2\}, \{1,2\},\{3\},\{1,3\},\{2,3\},\{1,2,3\},\{4\},\{1,4\},\ldots$
Among these we still have to define a relation $\le$ and it is perfectly fine to take the relation $\subseteq$ that works for set. The main pont is that it obeys the requird axioms for $\le$. So for example $x\subseteq y$ and $y\subseteq z$ imply $x\subseteq z$.
All in all, whart you gave as "naive reasoning is correct as $x\subseteq y$ and $y\subseteq $x$ implies $x=y$.
However, I don't know why specifically $U$ should be infinite and $X$ the set of finite subsets of $U$.
The argument works if $X$ is just any set of sets and $\le$ is $\subseteq$.
A: In answer to your second question, this is a major component on the axiomatic set theory. I believe the power theory states that if X is the set of finite subsets of a fixed infinite set U then ∀x∃y∀u(u∈y↔u⊆x). And there are many subsequent theories based on this axiom. http://mathworld.wolfram.com/AxiomofthePowerSet.html
