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

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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.