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

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