Proof of the De Morgan's laws under a σ−Algebra of events?

Question

Properties of $\mathscr{C}$, a $Boolean$ algebra:

1. $Ω∈\mathscr{C}$
2. If $A∈\mathscr{C}⇒A^c∈\mathscr{C}$
3. If $A,B∈\mathscr{C}⇒A∪B∈\mathscr{C}$

Demostrated properties of $\mathscr{C}$:

4. $∅∈\mathscr{C}$
5. If $A,B∈\mathscr{C}⇒A\cap B∈\mathscr{C}$
6. If $A_{1},...,A_{n}∈\mathscr{C}⇒\left(\bigcup_{i=1}^{i=n}A_{i}\right)∈\mathscr{C}$ and $\left(\bigcap_{i=1}^{i=n}A_{i}\right)∈\mathscr{C}$ with $n∈N^*$

Every $σ-Algebra$ has $(1)$, $(2)$, $(3)$ and the following:

7. If $A_{1},...,A_{n}∈\mathscr{C}⇒\left(\bigcup_{i=1}^{i=\infty}A_{i}\right)∈\mathscr{C}$

So, I want to prove that the following laws of $De$ $Morgan$ are fulfilled:

$$\left(\bigcap_{i=1}^{\infty}A_{i}\right)^c=\bigcup_{i=1}^{\infty}A^c_{i}$$

and

$$\left(\bigcup_{i=1}^{\infty}A_{i}\right)^c=\bigcap_{i=1}^{\infty}A^c_{i}$$

Can I say that...

• $X=\bigcap_{i=1}^{i=n}A_{i}$
• $Y=\bigcap_{i=n}^{\infty}A_{i}$
• $\left(\bigcap_{i=1}^{\infty}A_{i}\right)^c=\left( X \cap Y \right)^c$

To do this?

Let $A_{j} \in (X \cap Y)^c$. Then, $A_{j} \not\in X \cap Y$.

Because $X \cap Y = \{A_{k} | A_{k} \in X $ and $ A_{k} \in Y\}$, it must be the case that $A_{j} \not\in X$ or $A_{j} \not\in Y$.

If $A_{j} \not\in X$, then $A_{j} \in X^c$, so $A_{j} \in X^c \cup Y^c$.

Similarly, if $A_{j} \not\in Y$, then $A_{j} \in Y^c$, so $A_{j} \in X^c\cup Y^c$.

Thus, $\forall A_{j}( A_{j} \in (X\cap Y)^c \rightarrow A_{j} \in X^c \cup Y^c)$;

that is, $(X\cap Y)^c \subseteq X^c \cup Y^c$.

(...)

Answer 1

HINT: the proofs related to equality of sets (by example $A=B$) in general follow this strategy structured in two parts:

1. First we prove that $A\subseteq B$

2. After we prove the other direction, i.e. that $A\supseteq B$

Both statements together imply that $A=B$. Explicitly for the first part we choose some arbitrary point of $A$ and we show that it belong to $B$, hence $A\subseteq B$.

The other part is the same but reversed: we choose an arbitrary point of $B$ and we shows that it belong to $A$, hence $A\supseteq B$.

I think that with this information you can prove now the De Morgan laws for $\sigma$-algebras.

Answer 2

\begin{align} x \in \left(\bigcup_{i=1}^{\infty} A_i \right)^c &\iff \lnot \left(x\in \bigcup_{i=1}^{\infty} A_i \right) \\ &\iff \lnot ( \exists i\in[1..\infty), x \in A_i )\\ &\iff \forall i \in [1..\infty), x \not \in A_i \\ &\iff \forall i \in [1..\infty), x \in A_i^c \\ &\iff x \in \bigcap_{i=1}^\infty A_i^c \end{align}