structural:expansion
Preorder R\R

Every relation R {\displaystyle R} generates a preorder R ∖ R {\displaystyle R\backslash R} which is the left residual. In terms of converse and complements, R ∖ R ≡ R T R ¯ ¯. {\displaystyle R\backslash R\equiv {\overline {R^{\textsf {T}}{\bar {R}}}}.} Forming the diagonal of R T R ¯ {\displaystyle R^{\textsf {T}}{\bar {R}}}, the corresponding row of R T {\displaystyle R^{\textsf {T}}} and column of R ¯ {\displaystyle {\bar {R}}} will be of opposite logical values, so the diagonal is all zeros. Then

To show transitivity, one requires that (R ∖ R) (R ∖ R) ⊆ R ∖ R. {\displaystyle (R\backslash R)(R\backslash R)\subseteq R\backslash R.} Recall that X = R ∖ R {\displaystyle X=R\backslash R} is the largest relation such that R X ⊆ R. {\displaystyle RX\subseteq R.} Then

The inclusion relation Ω on the power set of U {\displaystyle U} can be obtained in this way from the membership relation ∈ {\displaystyle \in } on subsets of U {\displaystyle U}: