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# Math Formulas: Set Identities

### 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

 $$B \setminus A = B \setminus \left( A \cup B \right)$$
 $$B \setminus A = B \cap A'$$
 $$A \setminus A = \varnothing$$
 $$\left(A \setminus B \right) \cap C = \left(A \cap C \right) \setminus \left(B \cap C \right)$$
 $$A' = I \setminus A$$