Math formulas: Set identities

0 formulas included in custom cheat sheet

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

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

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

$$ A \setminus A = \varnothing $$

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

$$ A' = I \setminus A $$