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

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
