Editing Direct product (section)

== Direct product of binary relations ==
On the Cartesian product of two sets with [[binary relations]] <math>R \text{ and } S,</math> define <math>(a, b) T (c, d)</math> as <math>a R c \text{ and } b S d.</math> If <math>R \text{ and } S</math> are both [[reflexive relation|reflexive]], [[Irreflexive relation|irreflexive]], [[transitive relation|transitive]], [[Symmetric relation|symmetric]], or [[Antisymmetric relation|antisymmetric]], then <math>T</math> will be also.<ref>{{cite web| url = http://cr.yp.to/2005-261/bender1/EO.pdf| title = Equivalence and Order}}</ref> Similarly, [[total relation|totality]] of <math>T</math> is inherited from <math>R \text{ and } S.</math> If the properties are combined, that also applies for being a [[preorder]] and being an [[equivalence relation]]. However, if <math>R \text{ and } S</math> are [[connected relation|connected relations]], <math>T</math> need not be connected; for example, the direct product of <math>\,\leq\,</math> on <math>\N</math> with itself does not relate <math>(1, 2) \text{ and } (2, 1).</math>