Intersection

THEOREM    $$x \in A \cap B \Leftrightarrow x \in A \wedge x \in B$$
Int_iff: (?c $$\in$$ ?A $$\cap$$ ?B) = (?c $$\in$$ ?A $$\wedge$$ ?c $$\in$$ ?B)


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

THEOREM

The universal set is a left identity element of intersection.

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

Proof available

THEOREM

The universal set is a right identity element of intersection.

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

Proof available

THEOREM

Intersection is idempotent.

$$A \cap A = A$$

Proof available

THEOREM

Intersection is commutative.

$$A \cap B = B \cap A$$

Proof available

THEOREM

Intersection is associative.

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

Proof available

LEMMA

Intersection is left self-distributive.

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

Proof available

LEMMA

Intersection is right self-distributive.

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

Proof available

THEOREM

Intersection is self-distributive.

Proof available

THEOREM    $$S \cap T \subseteq S$$
Int_lower1: ?A $$\cap$$ ?B $$\subseteq$$ ?A


Proof available

THEOREM    $$S \cap T \subseteq T$$
Int_lower2: ?A $$\cap$$ ?B $$\subseteq$$ ?B


Proof available