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

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