# Union

THEOREM    $$x \in A \cup B \Leftrightarrow x \in A \vee x \in B$$
Un_iff: (?c $$\in$$ ?A $$\cup$$ ?B) = (?c $$\in$$ ?A $$\vee$$ ?c $$\in$$ ?B)


Proof unavailable

THEOREM

The empty set is a left identity element of union.

$$\emptyset \cup A = A$$

Proof available

THEOREM

The empty set is a right identity element of union.

$$A \cup \emptyset = A$$

Proof available

THEOREM

The universal set is a left zero element of union.

$$\mathcal{U} \cup A = \mathcal{U}$$

Proof available

THEOREM

The universal set is a right zero element of union.

$$A \cup \mathcal{U} = \mathcal{U}$$

Proof available

THEOREM

Union is idempotent.

$$A \cup A = A$$

Proof available

THEOREM

Union is commutative.

$$A \cup B = B \cup A$$

Proof available

THEOREM

Union is associative.

$$A \cup B \cup C = A \cup (B \cup C)$$

Proof available

LEMMA

Union is left self-distributive.

$$A \cup (B \cup C) = (A \cup B) \cup (A \cup C)$$

Proof available

LEMMA

Union is right self-distributive.

$$(B \cup C) \cup A = (B \cup A) \cup (C \cup A)$$

Proof available

THEOREM

Union is self-distributive.

Proof available

THEOREM    $$S \subseteq S \cup T$$

Proof available

THEOREM    $$T \subseteq S \cup T$$

Proof available