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

## 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:

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

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 $\left\langle{T_i}\right\rangle_{i \mathop \in I}$

## Proof

Suppose:

$\displaystyle x \in S \setminus \bigcup_{i \mathop \in I} T_i$

Note that by Set Difference is Subset we have that $x \in S$ (we need this later).

Then:

 $\displaystyle x$ $\in$ $\displaystyle S \setminus \bigcup_{i \mathop \in I} T_i$ $\displaystyle \iff \ \$ $\displaystyle x$ $\notin$ $\displaystyle \bigcup_{i \mathop \in I} T_i$ Definition of Set Difference $\displaystyle \iff \ \$ $\displaystyle \neg (\exists i \in I: (x$ $\in$ $\displaystyle T_i))$ Definition of Union of Family $\displaystyle \iff \ \$ $\displaystyle \forall i \in I: \neg (x$ $\in$ $\displaystyle T_i)$ De Morgan's Laws (Predicate Logic) $\displaystyle \iff \ \$ $\displaystyle \forall i \in I: x$ $\in$ $\displaystyle S \setminus T_i$ Definition of Set Difference: note $x \in S$ from above $\displaystyle \iff \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{i \mathop \in I} \left({S \setminus T_i}\right)$ Definition of Intersection of Family

Therefore:

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

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.