List of formula related to sets of identities
Definitions:
Universal set : Ι
Empty set: Φ
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
\[B \setminus A = B \setminus \left( A \cup B \right)\]
\[B \setminus A = B \cap A'\]
\[A \setminus A = \varnothing\]
\[\left(A \setminus B \right) \cap C = \left(A \cap C \right) \setminus \left(B \cap C \right)\]
\[A' = I \setminus A\]
