De Morgan's laws for sets

Statement

Let $U$ be a set and let $A$ and $B$ be subsets of $U$. Write $X^c = U \setminus X$ for the complement of a subset $X$ in $U$. Then $$(A \cup B)^{c} = A^{c} \cap B^{c},$$ $$(A \cap B)^{c} = A^{c} \cup B^{c}.$$

Remarks

The laws come from De Morgan's laws of logic, applied to $x \in A$ and $x \in B$. They hold also for arbitrary families: the complement of $\bigcup_i A_i$ is $\bigcap_i A_i^c$, and the complement of $\bigcap_i A_i$ is $\bigcup_i A_i^c$. With set differences, $C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B)$ for each set $C$.

Explanations

L1 human

An element is outside $A \cup B$ exactly when it is outside $A$ and outside $B$. An element is outside $A \cap B$ exactly when it is outside $A$ or outside $B$. A Venn diagram with two circles shows both laws.