There are three elements $a, b,$ and $c$.
How many sets are there of the form:
1) $\{\{x\}, \{y\}, \{x,y\}\}$, where $x, y \in \{a,b,c\}$
2) $\{\{x\}, \{x,y\}, \{x,z\}\}$, where $x, y, z \in \{a,b,c\}$
3) $\{\{x\}, \{y\}, \{x,y\}, \{x,z\}\}$, where $x, y, z \in \{a,b,c\}$
Attempt:
- I tried using combinations but was having a hard time.
- Say for the first example, there are three choices for $\{x\}$, then 2 choices for $\{y\}$, but since $x,y$ are now decided, is there a choice to make for $\{x,y\}$? Also since the order of these sets in the bigger set don't matter, how does this change things?
- Generating all the sets for this, I get that there are 3 distinct sets for the first example.