measure theory - A question about sigma-ring

How about choosing $B = B_1 - B_2$ and $C = C_1 -C_2$, we have $B in S(E cap A)$ by property of $S(E cap A)$, and $C in S(E)$, by property of $S(E)$.

The fact that this works can be seen by observing that both $B1 cup (C1−A)$, and $B2cup (C2−A)$ are in fact both disjoint unions. $B_1,B_2$ has everything to do with $A$ and $C_1 -A, C_2 -A$ has nothing to do with $A$ at all.

ps: I don't know why you use the and and or symbol here instead of intersection and union symbol.

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