In measure theory, they say that,
$\mathcal{F(C)}$ is a $\sigma$-algebra generated by C.
And,
(i) $\mathcal{F(C)} = \underset{\mathcal{C \subseteq P, \ P \ is \ a \ \sigma-algebra}}{\bigcap}$$\mathcal{P}$,
(ii) $\mathcal{F(C)}$ is the smallest $\sigma$-algebra containing C.
I'm having some conflicts in accepting it. Infact,
for example:
Let,
$X = \left\{ a,b,c,d,e \right\}$
$C = \left\{ a,b,c \right\}$
$\mathcal{F(C)} = \left\{\emptyset, X, \left\{ a \right\}, \left\{ b \right\}, \left\{ c \right\}, \left\{ a,b \right\}, \left\{ a,b,c \right\}, \left\{ a,c \right\}, \left\{ d,e \right\}, \left\{ a,b,d,e \right\}, \left\{ a,c,d,e \right\},\left\{ b,d,e\right\}, \left\{ c,d,e\right\}, \left\{ a,d,e \right\} , ...\right\}$
$\mathcal{F(\left\{ d, e \right\})} = \left\{ \emptyset, X, \left\{ d\right\}, \left\{ e\right\}, \left\{ d,e\right\}, \left\{a,b,c \right\}, \left\{ a,b,c,e \right\}, \left\{ a,b,c,d\right\} \right\}$
Clearly, $\mathcal{F(\left\{ d, e \right\})}$ contains less elements to that of $\mathcal{F(C)}$.
But how is that possible since $\mathcal{F(C)} = \underset{\mathcal{C \subseteq P, \ P \ is \ a \ \sigma-algebra}}{\bigcap}$$\mathcal{P}$ and $\mathcal{F(\left\{ d, e \right\})}$ is one of the $\mathcal{P}'s$?
Edit based on the comments of @ArturoMagidin:
$C = \left\{ \left\{a\right\},\left\{b\right\},\left\{c\right\} \right\}$ (since $C$ should be the collections of subsets of $X$)
$\mathcal{F(C)} = \left\{\emptyset, X, \left\{ a \right\}, \left\{ b \right\}, \left\{ c \right\}, \left\{ a,b \right\}, \left\{ a,b,c \right\}, \left\{ a,c \right\}, \left\{ d,e \right\}, \left\{ a,b,d,e \right\}, \left\{ a,c,d,e \right\},\left\{ b,d,e\right\}, \left\{ c,d,e\right\}, \left\{ a,d,e \right\} , ...\right\}$
$\mathcal{F(\left\{ \left\{d\right\},\left\{e\right\} \right\})} = \left\{ \emptyset, X, \left\{ d\right\}, \left\{ e\right\}, \left\{ d,e\right\}, \left\{a,b,c \right\}, \left\{ a,b,c,e \right\}, \left\{ a,b,c,d\right\} \right\}$
Since $\mathcal{F(\left\{ \left\{d\right\},\left\{e\right\} \right\})}$ doesn't contain $C$, it's not one of intersecting $P's$.