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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

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

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

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

$$ A \cup A = A $$ 
Commutativity

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

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

$$ A \cap A = A $$ 
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

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

$$ A \cap I = A $$ 
Complement of intersection and union

$$ A \cup A' = I $$ 

$$ A \cap A' = \varnothing $$ 
De Morgan's laws

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

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

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

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

$$ A \setminus A = \varnothing $$ 

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

$$ A' = I \setminus A $$ 
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