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I have been struggling to understand what makes a subset of a family of sets.

Definition: A set $A$ is a subset of another set $B$ if every element of the set $A$ is also in $B$.

For a family of sets $B$, are the elements of B referring to the sets which $B$ contains, or the elements inside each individual set. For example, say we are considering the family of sets $$A = \{\{1,2\}, \{3, 4\}, \{5, 6\}\}$$

If F = {3,5}, then is $F$ a subset of $A$? The whole set $F$ is not in $A$, but the elements of $F$ are.

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2 Answers 2

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Question What are the elements of $A$? Answer The sets $\{1,2\}, \{3,4\}, \{5,6\}$.

Second Question What are the elements of $F$? Answer The numbers $3, 5$.

Are all of the numbers $3,5$ part of $\{1,2\}, \{3,4\}, \{5,6\}$? The answer is no (none of those by the way). So $F$ is not a subset of $A$.

What you can say is that the elements of $F$ are elements of the elements of $A$.

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A subset of a family of sets is just a (smaller) family of sets, each of which is in the original family.

Using your example, if the family $\mathscr A=\{\{1,2\}, \{3,4\}, \{5,6\}\}$ (which is a family of three sets), then $F=\{3,5\}$ is not a subset of $\mathscr A$. Nor is it an element of $\mathscr A$, for that matter. It is really unrelated to $\mathscr A$.

Continuing the example, you can actually list all of the subsets of $\mathscr A$, each of which is a family of sets: $$\mathscr B_1=\{\}=\varnothing$$ $$\mathscr B_2=\{\{1,2\}\}$$ $$\mathscr B_3=\{\{3,4\}\}$$ $$\mathscr B_4=\{\{5,6\}\}$$ $$\mathscr B_5=\{\{1,2\}, \{3,4\}\}$$ $$\mathscr B_6=\{\{1,2\}, \{5,6\}\}$$ $$\mathscr B_7=\{\{3,4\}, \{5,6\}\}$$ $$\mathscr B_8=\{\{1,2\},\{3,4\}, \{5,6\}\}=\mathscr A$$

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