Knowlpedia

Supremum and Infimum Algebra

How supremum and infimum interact with basic set operations such as translation and scaling.

Supremum/infimum algebra: Let A,B⊆RA,B \subseteq \mathbb{R} be nonempty . Define A+B={a+b: a∈A, b∈B}A+B=\{a+b:\ a\in A,\ b\in B\} and, for c∈Rc\in\mathbb{R}, cA={ca: a∈A}cA=\{ca:\ a\in A\}. Assume the displayed quantities are finite. Then:

  • sup⁡(A+B)=sup⁡(A)+sup⁡(B)\sup(A+B)=\sup(A)+\sup(B) and inf⁡(A+B)=inf⁡(A)+inf⁡(B)\inf(A+B)=\inf(A)+\inf(B).
  • If c≥0c\ge 0, then sup⁡(cA)=c sup⁡(A)\sup(cA)=c\,\sup(A) and inf⁡(cA)=c inf⁡(A)\inf(cA)=c\,\inf(A).
  • If c≤0c\le 0, then sup⁡(cA)=c inf⁡(A)\sup(cA)=c\,\inf(A) and inf⁡(cA)=c sup⁡(A)\inf(cA)=c\,\sup(A).
  • Writing −A={−a: a∈A}-A=\{-a:\ a\in A\}, one has sup⁡(−A)=−inf⁡(A)\sup(-A)=-\inf(A) and inf⁡(−A)=−sup⁡(A)\inf(-A)=-\sup(A).
Remarks

Finiteness follows, for example, from the relevant or hypotheses. These rules are frequently paired with the to turn statements about and into ε\varepsilon-estimates.