Please help me to show analytically that for $A\subseteq Ω$, the following collection of sets $F =\{\emptyset,\Omega,A,A^c\}$, is a $\sigma$-algebra of subsets of $\Omega$.
this is the first time trying this so please help me. I'm really struggling with this matter.
I`m not sure about the tag
A $\sigma$ algebra $F$ of subsets of $\Omega$ fulfills the following 4 conditions:
1) $\Omega \in F$ (this is fulfilled by your definition of F)
2) $X \in F \implies X^c \in F$ (which is also trivially fulfilled by your definition)
3) $(X_n: n \in \mathbb{N}) \in F \implies \bigcup_{n\in \mathbb{N}} X_n \in F$ (this is also fulfilled as the only subsets aside from $\Omega$ and $\emptyset$ is the disjoint pair of $A, A^c$).
4) Finite intersections which is implied by (3).
And so, $F$ is a $\sigma$ algebra of subsets of $\Omega$.