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

## Theorem

Let $S$ and $T$ be sets.

Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.

Then:

#### Difference with Intersection

$\ds S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$

where:

$\ds \bigcup_{i \mathop \in I} T_i := \set {x: \exists i \in I: x \in T_i}$

that is, the union of $\family {T_i}_{i \mathop \in I}$.

#### Difference with Union

$\ds S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$

where:

$\ds \bigcap_{i \mathop \in I} T_i := \set {x: \forall i \in I: x \in T_i}$

that is, the intersection of $\family {T_i}_{i \mathop \in I}$.

## Source of Name

This entry was named for Augustus De Morgan.