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.
Best Answer
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: