1. ## Family of sets

$\mathcal{F}=\{\{1,2\}, \{3,4\},\{5,6\}\}$

$\mathcal{F}$ is a family of sets.

Question: Is $\{\{2,8\}\}$ a family of set(s)? If not, is there a name for it?

2. 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.

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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.