De Morgan's Laws (Set Theory)/Set Difference/Family of Sets

Theorem

Let $S$ and $T$ be sets.

Let $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ be a family of subsets of $T$.

Then:

Difference with Intersection

$\displaystyle S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \left({S \setminus T_i}\right)$

where:

$\displaystyle \bigcup_{i \mathop \in I} T_i := \left\{{x: \exists i \in I: x \in T_i}\right\}$

that is, the Definition:Union of Family of $\left\langle{T_i}\right\rangle_{i \mathop \in I}$.

Difference with Union

$\displaystyle S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \left({S \setminus T_i}\right)$

where:

$\displaystyle \bigcap_{i \mathop \in I} T_i := \left\{{x: \forall i \in I: x \in T_i}\right\}$

that is, the intersection of $\left\langle{T_i}\right\rangle_{i \mathop \in I}$

Source of Name

This entry was named for Augustus De Morgan.