Supremum and Infimum

Topics: Discrete Mathematics - poset

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(definition)

Let $(X,⪯)$ be a poset and let $B\subseteq X$

• The supremum of $B$ (${\sup }_x(B)$) is the minimum upper bound of $B$
• The infimum of $B$ (${\inf }_x(B)$) is the maximum lower bound of $B$

Not Necessarily in the Set

The supremum and the infimum of a given set aren’t necessarily in that set.

The supremum and infimum help us define lattices.