Math formulas: Set identities

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Definitions:

Universal set : $I$

Empty set: $\varnothing$

Union of sets

$$A \cup B = \left\{x : x \in A ~~ or ~~ x \in B \right\}$$

Intersection of sets

$$A \cap B = \left\{x : x \in A ~~ and ~~ x \in B \right\}$$

Complement

$$A' = \left\{ x \in I : x \not \in A \right\}$$

Difference of sets

$$A \setminus B = \left\{x : x \in A ~~ and ~~ x \not \in B \right\}$$

Cartesian product

$$A \times B = \left\{ (x,y) : x \in A ~~ and ~~ y \in B \right\}$$

Set identities involving union

Commutativity

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

Associativity

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

Idempotency

$$A \cup A = A$$

Set identities involving intersection

Commutativity

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

Associativity

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

Idempotency

$$A \cap A = A$$

Set identities involving union and intersection

Distributivity

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

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

Domination

$$A \cap \varnothing = \varnothing$$

$$A \cup I = I$$

Identity

$$A \cup \varnothing = \varnothing$$

$$A \cap I = A$$

Set identities involving union, intersection and complement

Complement of intersection and union

$$A \cup A' = I$$

$$A \cap A' = \varnothing$$

De Morgan's laws

$$\left( A \cup B \right)' = A' \cap B~'$$

$$\left(A \cap B \right)' = A' \cup B~'$$

Set identities involving difference

$$B \setminus A = B \setminus \left( A \cup B \right)$$

$$B \setminus A = B \cap A'$$

$$A \setminus A = \varnothing$$

$$\left(A \setminus B \right) \cap C = \left(A \cap C \right) \setminus \left(B \cap C \right)$$

$$A' = I \setminus A$$