# de Morgan's laws on generalised union and intersection.

• Sep 22nd 2012, 06:00 AM
sunmalus
de Morgan's laws on generalised union and intersection.
Hi, I need to prove de Morgan's laws for generalised union and intersections, I came up with a proof but I don't know if it's works with what I know.
Here is what I know:
Given A, a set and B such that $\displaystyle B \subseteq A$ then $\displaystyle B^{c}=A-B$.
$\displaystyle \mathcal{A}$ is a collection of sets.
$\displaystyle \bigcup_{A \in \mathcal{A}}A=\{ x| \exits A \in \mathcal {A} \text{ such that } x \in A \}$.
$\displaystyle \bigcap_{A \in \mathcal{A}}A= \{x| x \in A, \forall A \in \mathcal A\}$.

proof:
$\displaystyle \text{1. }(\bigcup_{A \in \mathcal{A}}A)^{c}=\{x| x \notin A, \forall A \in \mathcal{A}\}=\{x|x \in A^{c}, \forall A \in \mathcal{A}\}=\bigcap_{A \in \mathcal{A}}A^{c}$.
$\displaystyle \text{2. }(\bigcap_{A \in \mathcal{A}}A)^{c}=\{x| \exits A \in \mathcal{A} \text{ such that } x \notin A\}= \{ x| \exists A \in \mathcal{A} \text{ such that } x \in A^{c}\}=\bigcup_{A \in \mathcal{A}}A^c$.

I wanted to make a remark about the lack of a quantifier over A in the definition of union, but then I saw that it is a LaTeX error. Since you put existential quantifier in the beginning, I would do the same with the universal quantifier. I would also add an extra equality $\displaystyle \left(\bigcup_{A \in \mathcal{A}}A\right)^{c}=\{x\mid\neg\exists A\in\mathcal{A}.\,x\in A\}$ for clarity, and similarly for intersection.