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Infimum and Supremum

A value x is a infimum of S, if it is the greatest lower bound of S:

x is infimum of S w.r.t. <= : <=>

x is lower bound of S w.r.t. <= /\

forall y: y is lower bound of S w.r.t. <= => y <= x.

A value x is a supremum of S, if it is the least upper bound of S:

x is supremum of S w.r.t. <= : <=>

x is upper bound of S w.r.t. <= /\

forall y: y is upper bound of S w.r.t. <= => x <= y.

Infimum respectively supremum of a set S need not be element of S.