De Morgan's Laws (Set Theory)/Set Difference

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Theorem

Let $S, T_1, T_2$ be sets.

Let:

$T_1 \cap T_2$ denote set intersection
$T_1 \cup T_2$ denote set union.

Then:

Difference with Intersection

$S \setminus \left({T_1 \cap T_2}\right) = \left({S \setminus T_1}\right) \cup \left({S \setminus T_2}\right)$


Difference with Union

$S \setminus \left({T_1 \cup T_2}\right) = \left({S \setminus T_1}\right) \cap \left({S \setminus T_2}\right)$


DeMorganMinusIntersection.png DeMorganMinusUnion.png

General Case

Let $S$ and $T$ be sets.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.


Then:

Difference with Intersection

$\displaystyle S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \left({S \setminus T'}\right)$

where:

$\displaystyle \bigcap \mathbb T := \left\{{x: \forall T' \in \mathbb T: x \in T'}\right\}$

that is, the intersection of $\mathbb T$


Difference with Union

$\displaystyle S \setminus \bigcup \mathbb T = \bigcap_{T' \mathop \in \mathbb T} \left({S \setminus T'}\right)$

where:

$\displaystyle \bigcup \mathbb T := \left\{{x: \exists T' \in \mathbb T: x \in T'}\right\}$

that is, the union of $\mathbb T$.


Family of Sets

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 union 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.