Properties of Set Operations

These are the algebraic laws that union, intersection and complement obey (all quantifiers are "for all sets $A,B,C$ inside a fixed universal set $U$"):

Commutative: $A\cup B=B\cup A$; $A\cap B=B\cap A$.

Associative: $(A\cup B)\cup C=A\cup(B\cup C)$; $(A\cap B)\cap C=A\cap(B\cap C)$.

Distributive: $A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$; $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$.

Identity: $A\cup\varnothing=A$; $A\cap U=A$.

Idempotent: $A\cup A=A$; $A\cap A=A$.

Absorption: $A\cup(A\cap B)=A$; $A\cap(A\cup B)=A$.

De Morgan's Laws:

On symmetric difference: $A\,\Delta\,B=B\,\Delta\,A$ (commutative); $(A\,\Delta\,B)\,\Delta\,C=A\,\Delta\,(B\,\Delta\,C)$ (associative); $A\cap(B\,\Delta\,C)=(A\cap B)\,\Delta\,(A\cap C)$ (intersection distributes over symmetric difference).

On $\varnothing$ and $U$: $\varnothing'=U$; $U'=\varnothing$; $A\cup A'=U$; $A\cap A'=\varnothing$; $A\cup U=U$; $A\cap U=A$.

Cardinality (inclusion-exclusion). For finite sets:

if $A,B$ are disjoint, $n(A\cup B)=n(A)+n(B)$. For three finite sets, …