# Set Identities Formulas

## Definitions:

Universal set: I
Empty set: $$\emptyset$$

Union of sets

$$A \cup B = \{x|x \in A or x \in B\}$$

Intersection of sets

$$A \cap B = \{x|x \in A and x \in B\}$$

Complement

$$A’ = \{ x \in l | x \in A \}$$

Difference of sets

$$B \backslash A = \{ x | x \in B and x \notin A \}$$

Cartesian product

$$A x B = \{ ( x, y ) | x \in A and y \in B \}$$

## Set identities involving union

Commutativity

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

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

$$A \cup A = A$$

## Set Identities involving intersection

Commutativity

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

Associativity

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

$$A \cap A = A$$

## Set identities involving union and intersection

Distributivity

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

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

Domination

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

$$A \cup l = l$$

Identity

$$A \cup \emptyset = A$$

$$A \cap l = A$$

## Set identities involving union, intersection and complement

Complement of intersection and union

$$A \cup A’ = l$$

$$A \cap A’ = \emptyset$$

De Morgan’s laws

$$(A \cup B)’ = A’ \cap B’$$

$$(A \cap B)’ = A’ \cup B’$$

Set identities involving difference

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

$$B \backslash A = B \cap A’$$

$$A \backslash A = \emptyset$$

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

$$A’ = l \backslash A$$