Notation for defining cover of a set

Question

Is it appropriate to define a cover $C$ of non-singleton subsets of $S$ using the following notation? $$ C := \left\{ X \subseteq S: \vert X \vert \ge 2 \ \text{and} \ \bigcup_{X \in C} X = S \right\}, $$ where $\vert \cdot \vert$ denotes the cardinality of a set.

In particular, I am annoyed by the presence of $C$ in the left- as well as in the right-hand side.

Answer 1

When using set-builder notation, the first half of the notation specifies a dummy variable used to denote elements of the set, and the second half of the notation specifies properties which elements of the set must have. In the notation $$C := \left\{ X \subseteq S: \vert X \vert \ge 2 \ \text{and} \ \bigcup_{X \in C} X = S \right\},$$ the expression $\bigcup_{X\in C} X = S$ does not specify properties of the elements $X$, but rather attempts to specify a property of the set $C$ itself. This is bound to cause problems. Even if we eliminate this expression, I don't think that the notation expresses what you want. Specifically, the expression $$ C := \left\{ X \subseteq S: \vert X \vert \ge 2\right\} $$ is the collection of all non-empty, non-singleton subsets of $S$. This happens to be a cover, but is a very specific cover. If it is what you are actually meaning to express, then the union is redundant. Otherwise, the notation is not the notation you want.

Instead, I would write something like the following:

Let $S$ be a set. A cover of $S$ is a collection $\mathscr{C} \in \mathscr{P}(X)$ such that

$$ S \subseteq \bigcup_{C \in \mathscr{C}} C. $$

Fix a cover $\mathscr{C}$ of $S$ such that $C \in \mathscr{C}$ implies that $|C| \ge 2$.

Answer 2

Using $s$ for both the dummy variable in the definition of the set and the dummy variable in the union is indeed a problem. The union itself is also problematic: $S$ is $\{s:s\in S\}$, not $\bigcup\{s:s\in S\}$. Finally, even if the definition said what I think you intend, it wouldn’t define anything: there is nothing in it that picks out any specific cover of $S$. (That does change if you simply delete the problematic union: then $C$ is defined to be the collection of all subsets of $S$ having at least two elements.)

If you want to define a cover of $S$, you must actually specify how its members are chosen. If you simply want to say that $C$ is a cover of $S$ with certain properties, you can do that, but you won’t be defining $C$. For instance, you can say ‘Let $C$ be a cover of $S$ such that $|c|\ge 2$ for each $c\in C$.’