Which of these quantity systems corresponds to sigma algebra? Given is the set $Ω = \left\{ 3,4,5,6,7\right\}  $.
 
Determine which of the given quantity systems corresponds to an sigma algebra and justify it.
$1. \left\{\left\{\right\},\left\{3\right\},\left\{5\right\},\left\{3,4\right\},\left\{3,4,6,7\right\},\left\{3,4,5,6,7\right\}\right\}$
$2. \left\{\left\{\right\},\left\{5\right\},\left\{3,4,6,7\right\},\left\{3,4,5,6,7\right\}\right\}$
$3. \left\{\left\{\right\},\left\{3\right\},\left\{4\right\},\left\{5\right\},\left\{6\right\},\left\{7\right\},\left\{3,4\right\},\left\{3,5\right\},\left\{3,6\right\},\left\{3,7\right\},\left\{4,5\right\},\left\{4,6\right\},\left\{4,7\right\},\left\{5,6\right\},\left\{5,7\right\},\left\{6,7\right\},\left\{3,4,5\right\},\left\{3,4,6\right\},\left\{3,4,7\right\},\left\{3,5,6\right\},\left\{3,5,7\right\},\left\{3,6,7\right\},\left\{4,5,6\right\},\left\{4,5,7\right\},\left\{4,6,7\right\},\left\{5,6,7\right\},\left\{3,4,5,6\right\},\left\{3,4,5,7\right\},\left\{3,4,6,7\right\},\left\{3,5,6,7\right\},\left\{4,5,6,7\right\},\left\{3,4,5,6,7\right\}\right\}$
$4. \left\{\left\{\right\},\left\{5\right\}\right\}$
I think that 1. and 3. are corresponded. Is that correct and if yes how to justify that?

*

*is not a $\sigma$-algebra because $\{3\}\cup \{5\}=\{3,5\}$ is not in 1.

*is a $\sigma$-algebra because $\{\}$, $\Omega$, and all the possible intersections and unions of the elements of 2. are in 2.

*works because all the $2^{\Omega}=32$ subsets (of $\Omega $) are in 3.

*would not work because $\Omega$ is not in 4.

