Principle of duality

Statement

Each of the identities stated above is one of a pair of identities such that each can be transformed into the other by interchanging $\cup$ and $\cap$, while also interchanging $\varnothing$ and $\boldsymbol{U}$.

These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions and intersections, interchanging $\boldsymbol{U}$ and $\varnothing$ and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual.