Criterion for a subset lie inside a sigma algebra

Question

We know that $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a subset of a power sets of $X$ satisfying

(i) $\emptyset,X\in\mathcal{A}$

(ii) $A\in\mathcal{A}$ then $X-A\in \mathcal{A}$.

(iii) $A_k\in\mathcal{A}$ then $\bigcup_{k=1}^\infty A_k,\;\bigcap_{k=1}^\infty A_k\in\mathcal{A}$

Now comes a very stupid question. What is the criteria for a subset $B\subset X$ lies in $\mathcal{A}$? Is a subset satisfying (ii) sufficient?

Answer 1

Yes a set $B\subseteq X$ is in $\mathcal{A}$ if $B=X\setminus A$ for some $A\in\mathcal{A}$ (ii). Additionally, $B\in\mathcal{A}$ if $B$ is a countable union or intersection of sets in $\mathcal{A}$ (iii).