What are the major properties of indexed unions and intersections?

Question

I'm pretty confident reasoning about expressions involving the following operations.

• binary unions
• binary intersections
• complementation

However, I am less confident in my ability to reason about expressions that also involve:

• indexed unions, and/or
• indexed intersections.

What are the major laws that apply here?

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To get the ball rolling, here's 5 that I know of.

Notation. Let $\Omega$ denote a set. Define $A^c = \Omega \setminus A$ for all $A \subseteq \Omega.$ Suppose furthermore that $X$ and $Y$ are functions $I,J \rightarrow \mathcal{P}(\Omega),$ where $I$ and $J$ are just some arbitrary index sets. We will use the $X_i$ in preference to $X(i)$. Lastly, define that the empty intersection equals $\Omega.$ That is, $$\bigcap_{i \in \emptyset} X_i = \Omega.$$

Then we have the following.

1. If $\mathrm{ran}(X) \subseteq \mathrm{ran}(Y)$, then $$\bigcup_{i \in I}X_i \subseteq \bigcup_{j \in J}Y_j, \quad \bigcap_{i \in I}X_i \supseteq \bigcap_{j \in J}Y_j$$

2. (Corollary). If $\mathrm{ran}(X) = \mathrm{ran}(Y),$ then $$\bigcup_{i \in I}X_i = \bigcup_{j \in J}Y_j, \quad \bigcap_{i \in I}X_i = \bigcap_{j \in J}Y_j$$

3. Distributivity Laws. For all $A \in \mathcal{P}(\Omega)$ we have $$A \cap \bigcup_{i \in I}X_i = \bigcup_{i \in I}A \cap X_i,\quad A \cup \bigcap_{i \in I}X_i = \bigcap_{i \in I}A \cup X_i.$$

4. De Morgan's Laws $$\left(\bigcup_{i \in I}X_i\right)^c = \bigcap_{i \in I}X_i^c,\quad \left(\bigcap_{i \in I}X_i\right)^c = \bigcup_{i \in I}X_i^c$$

5. If $Z : K \rightarrow \mathcal{P}(\Omega)$ is a function with the property that $\mathrm{ran}(Z) = \mathrm{ran}(X) \cup \mathrm{ran}(Y)$, then

$$\bigcup_{k \in K}Z_k = \left(\bigcup_{i \in I}X_i\right) \cup \left(\bigcup_{j \in J}Y_j\right).$$

Okay, but what else? What major laws are missing from the above list?

Answer 1

What about things like if $K\subset I$ then $\bigcup_{k \in K} X_k \subset \bigcup_{i \in I} X_i\,?$