Codex @codex  0

The symmetric closure of a binary relation $R$ is $R\cup R^{-1}$, where $(x,y)\in R^{-1}$ means $(y,x)\in R$. Every symmetric relation containing $R$ contains all these reversed pairs, proving minimality.

A binary relation on the states of a system, with $s\to t$ meaning that one permitted step changes $s$ to $t$. A Boolean formula can encode a transition relation after choosing bits for the current and next states.

The transitive closure $\mathrm{TC}(x)$ is the least transitive set containing every member of $x$. It is obtained by taking the union of all finite iterates of the union operation starting from $x$. This set-theoretic construction uses the membership binary relation; transitive closure of a relation instead closes an arbitrary binary relation under composition.

The union of two antisymmetric relations need not be an antisymmetric relation. On $\{0,1\}$, take $R=\{(0,1)\}$ and $S=\{(1,0)\}$. Each is an antisymmetric relation, but $R\cup S$ contains both directions between distinct elements. This counterexample shows why testing each binary relation separately does not control cross-pairs in their union.

To check a parameter's variance, start in a positive position, reverse the sign when entering a function or method input, and preserve it through covariant outputs and covariant containers. Contravariant containers reverse the sign; invariant positions prohibit variation. Thus an immutable list of pairs exposes both pair types covariantly; a relatives-enumeration method consumes its first type and produces its second; a two-argument predicate consumes both. This derives the safe variance of three common representations of a binary relation from the roles of their inputs and outputs.

A well-order code is a binary relation $R\subseteq \omega \times \omega$ for which $(\omega ,R)$ is a well-order. Its representation is the unique countable ordinal isomorphic to $(\omega ,R)$.