Properties of Sets

Laws

All sets referred to below are subsets of a 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

$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$ $A\cap (B\cup C)=(A\cap B)\cup (B\cap C)$

Identity

$A\cup \varnothing =A$ $A\cap U=A$

Complement

$A\cup A^c=U$ $A\cap A^c=\varnothing$

Double Complement

$(A^c{)}^c=A$

Idempotent

$A\cup A=A$ $A\cap A=A$

Universal Bound

$A\cup U=U$ $A\cap \varnothing =\varnothing$

De Morgan’s

$(A\cup B{)}^c=A^c\cap B^c$ $(A\cap B{)}^c=A^c\cup B^c$

Absorption

$A\cup (A\cap B)=A$ $A\cap (A\cup B)=A$

Complements of $U$ and $\varnothing$

$U^c=\varnothing$ ${\varnothing }^c=U$

Set Difference

$A-B=A\cap B^c$

Subset Relations

Inclusion of Intersection

$A\cap B\subseteq A$ $A\cap B\subseteq B$

Inclusion in Union

$A\subseteq A\cup B$ $B\subseteq A\cup B$

Transitive Property of Subsets

$A\subseteq B\wedge B\subseteq C\to A\subseteq C$