Exercise 574Logic, sets and proofs › Sets and mapsHigh schoolAdded 3 Nov 2026

Complement laws

#complement#De Morgan#laws

Statement

Prove that $\overline{A\cup B}=\overline A\cap\overline B$ and $\overline{A\cap B}=\overline A\cup\overline B$ .

Hint

Use element-wise logic.

Show solutionHide solution

First law. $x\in\overline{A\cup B}\iff x\notin A\cup B\iff x\notin A$ and $x\notin B\iff x\in\overline A\cap\overline B$ .

Second law. $x\in\overline{A\cap B}\iff$ not ( $x\in A$ and $x\in B$ ) $\iff x\notin A$ or $x\notin B\iff x\in\overline A\cup\overline B$ .

Conclusion. Both De Morgan laws hold.