linguistic:translation, structural:expansion, quality:corruption, structural:reformatting
Suprema

In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete.

One basic property of the supremum is sup { f (t) + g (t): t ∈ A } ≤ sup { f (t): t ∈ A } + sup { g (t): t ∈ A } {\displaystyle \sup\{f(t)+g(t):t\in A\}~\leq ~\sup\{f(t):t\in A\}+\sup\{g(t):t\in A\}} for any functionals f {\displaystyle f} and g. {\displaystyle g.}

The supremum of a subset S {\displaystyle S} of (N, ∣) {\displaystyle (\mathbb {N},\mid \,)} where ∣ {\displaystyle \,\mid \,} denotes " divides ", is the lowest common multiple of the elements of S. {\displaystyle S.}

The supremum of a set S {\displaystyle S} containing subsets of some set X {\displaystyle X} is the union of the subsets when considering the partially ordered set (P (X), ⊆) {\displaystyle (P(X),\subseteq)}, where P {\displaystyle P} is the power set of X {\displaystyle X} and ⊆ {\displaystyle \,\subseteq \,} is subset.