I think that it is not a $\sigma$-algebra of $\{1, 2, 3\},$ because for example
- $\{1, 2\} \cap \{1, 3\} = \{1\}$
- $\{1, 3\} \cap \{2, 3\} = \{3\}$
- $\{1, 2\}' = \{3\},$ etc.
which are not elements in $F.$ However, I'm not sure if this is right, because I'm not sure if I understand what a $\sigma$-algebra is; I haven't heard it explained thoroughly.