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

$S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$

Difference with Union

$S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$

General Case

Let $S$ and $T$ be sets.

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T \subseteq \powerset T$.

Then:

Difference with Intersection

$\ds S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$

where:

$\ds \bigcap \mathbb T := \set {x: \forall T' \in \mathbb T: x \in T'}$

that is, the intersection of $\mathbb T$

Difference with Union

$\ds S \setminus \bigcup \mathbb T = \bigcap_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$

where:

$\ds \bigcup \mathbb T := \set {x: \exists T' \in \mathbb T: x \in T'}$

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

Family of Sets

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.