Set identities

We now consider the basic set identities that relate the various set operations.

The sets $A,$ $B,$ $C$ below are subsets of a universal set $U.$

Identity Laws

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

Domination Laws

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

Idempotent Laws

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

Complement Laws

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

Double Complement Law

$${\left(A^c\right)}^c=A$$

Commutative Laws

$$A\cup B=B\cup A,A\cap B=B\cap A$$

Associative Laws

$$A\cup \left(B\cup C\right)=\left(A\cup B\right)\cup C,A\cap \left(B\cap C\right)=\left(A\cap B\right)\cap C$$

Distributive Laws

$$A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right),A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$$

De Morgan's Laws

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

Absorption Laws

$$A\cup \left(A\cap B\right)=A,A\cap \left(A\cup B\right)=A$$

Complements of $U$ and $\varnothing$

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

Set Difference Law

$$A\mathrm{\setminus }B=A\cap B^c$$