Set Complement

Complementary relation

A binary relation${\displaystyle R}$ is defined as a subset of a product of sets${\displaystyle X\times Y.}$ The complementary relation${\displaystyle {\bar {R}}}$ is the set complement of ${\displaystyle R}$ in ${\displaystyle X\times Y.}$ The complement of relation ${\displaystyle R}$ can be written $${\displaystyle {\bar {R}}\ =\ (X\times Y)\setminus R.}$$ Here, ${\displaystyle R}$ is often viewed as a logical matrix with rows representing the elements of ${\displaystyle X,}$ and columns elements of ${\displaystyle Y.}$ The truth of ${\displaystyle aRb}$ corresponds to 1 in row ${\displaystyle a,}$ column ${\displaystyle b.}$ Producing the complementary relation to ${\displaystyle R}$ then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.