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

Parent topics