Empty set


Definition

$$\emptyset = \{x : \bot\}$$
THEOREM   

The empty set is unique.

Proof available

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 empty set is a subset of every set.

$$\emptyset \subseteq A$$

Proof available

LEMMA    \(A \times \emptyset = \emptyset\)

Proof unavailable

LEMMA    \(\emptyset \times A = \emptyset\)

Proof unavailable

THEOREM   

The empty set is a left zero element of intersection.

$$\emptyset \cap A = \emptyset$$

Proof available

THEOREM   

The empty set is a right zero element of intersection.

$$A \cap \emptyset = \emptyset$$

Proof available

Proper supersets