# De Morgan's Laws (Set Theory)/Proof by Induction

## Theorem

Let $\mathbb T = \left\{{T_i: i \mathop \in I}\right\}$, where each $T_i$ is a set and $I$ is some finite indexing set.

Then:

### Difference with Intersection

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

### Difference with Union

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

## Source of Name

This entry was named for Augustus De Morgan.