$\displaystyle \mathcal{F}=\{\{1,2\}, \{3,4\},\{5,6\}\}$
$\displaystyle \mathcal{F}$ is a family of sets.
Question: Is $\displaystyle \{\{2,8\}\}$ a family of set(s)? If not, is there a name for it?
What's your definition of a "family of sets"? Is it just a "set of sets"? If the answer is 'yes', then of course {{2 8}} is a family of sets.
/
Sometimes, though, "family" has a more special meaning, which is "function" where the domain is called "the index set" and the range is called "the indexed set" [refer to Halmos's 'Naive Set Theory']. But, with that definition, your F is not a family.