Editing Infimum and supremum (section)

== Duality ==

If one denotes by <math>P^{\operatorname{op}}</math> the partially-ordered set <math>P</math> with the [[Converse relation|opposite order relation]]; that is, for all <math>x \text{ and } y,</math> declare: 
<math display=block>x \leq y \text{ in } P^{\operatorname{op}} \quad \text{ if and only if } \quad x \geq y \text{ in } P,</math>
then infimum of a subset <math>S</math> in <math>P</math> equals the supremum of <math>S</math> in <math>P^{\operatorname{op}}</math> and vice versa.

For subsets of the real numbers, another kind of duality holds: <math>\inf S = - \sup (- S),</math> where <math>-S := \{ -s ~:~ s \in S \}.</math>