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Algebraic properties of sup and inf

Supremum and infimum behave predictably under inclusion, translation, scaling, and unions

Let E,F⊆RE,F\subseteq\mathbb R be nonempty and bounded above or below as required, and let c∈Rc\in\mathbb R.

Order and translation
  • If E⊆FE\subseteq F, then sup⁡E≤sup⁡F\sup E\le\sup F when both sets are bounded above, and inf⁡E≥inf⁡F\inf E\ge\inf F when both are bounded below.
  • For E+c={x+c:x∈E}E+c=\{x+c:x\in E\},
    sup⁡(E+c)=sup⁡E+c,inf⁡(E+c)=inf⁡E+c.\sup(E+c)=\sup E+c,\qquad\inf(E+c)=\inf E+c.
Scaling

For λE={λx:x∈E}\lambda E=\{\lambda x:x\in E\}:

  • If λ≥0\lambda\ge0, then
    sup⁡(λE)=λ sup⁡E,inf⁡(λE)=λ inf⁡E.\sup(\lambda E)=\lambda\,\sup E,\qquad \inf(\lambda E)=\lambda\,\inf E.
  • If λ<0\lambda<0, then
    sup⁡(λE)=λ inf⁡E,inf⁡(λE)=λ sup⁡E.\sup(\lambda E)=\lambda\,\inf E,\qquad \inf(\lambda E)=\lambda\,\sup E.

In particular,

sup⁡(−E)=−inf⁡E,inf⁡(−E)=−sup⁡E.\sup(-E)=-\inf E,\qquad \inf(-E)=-\sup E.
Finite unions

If EE and FF are bounded above, then

sup⁡(E∪F)=max⁡{sup⁡E,sup⁡F}.\sup(E\cup F)=\max\{\sup E,\sup F\}.

If they are bounded below, then

inf⁡(E∪F)=min⁡{inf⁡E,inf⁡F}.\inf(E\cup F)=\min\{\inf E,\inf F\}.