Common Set Laws

Idempotent Laws

$$\begin{array}{rl}A\cup A& =A\\ A\cap A& =A\end{array}$$

Commutative Laws

$$\begin{array}{rl}A\cup B& =B\cup A\\ A\cap B& =B\cap A\end{array}$$

Associative Laws

$$\begin{array}{rl}(A\cup B)\cup C& =A\cup (B\cup C)\\ (A\cap B)\cap C& =A\cap (B\cap C)\end{array}$$

Distributive Laws

$$\begin{array}{rl}A\cup (B\cap C)& =(A\cup B)\cap (A\cup C)\\ A\cap (B\cup C)& =(A\cap B)\cup (A\cap C)\end{array}$$

Identity Laws

$$\begin{array}{rl}A\cup \varnothing & =A\\ A\cap U& =A\end{array}$$

Domination Laws

$$\begin{array}{rl}A\cup U& =U\\ A\cap \varnothing & =\varnothing \end{array}$$

Complement Laws

$$\begin{array}{rl}A\cup A^C& =U\\ A\cap A^C& =\varnothing \\ (A^C{)}^C& =A\\ U^C& =\varnothing \\ {\varnothing }^C& =U\end{array}$$

De Morgan’s Laws

$$\begin{array}{rl}(A\cup B{)}^C& =A^C\cap B^C\\ (A\cap B{)}^C& =A^C\cup B^C\end{array}$$

Absorption Laws

$$\begin{array}{rl}A\cup (A\cap B)& =A\\ A\cap (A\cup B)& =A\end{array}$$

Set Difference Laws

$$\begin{array}{rl}A-(B\cup C)& =(A-B)\cap (A-C)\\ A-(B\cap C)& =(A-B)\cup (A-C)\\ A-B& =A\cap B^C\end{array}$$