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Bounds and Extrema

  • Bounds and Extrema #Card
    • For an ordered set $(X, \le)$ and $A \subset X$: - Upper Bounds: $A^+ := {x \in X : \forall a \in A, a \le x}$ - Lower Bounds: $A^- := {x \in X : \forall a \in A, a \ge x}$ - Supremum (Least Upper Bound): The least element of $A^+$, denoted by $\bigvee A$ or $\sup A$. - Infimum (Greatest Lower Bound): The greatest element of $A^-$, denoted by $\bigwedge A$ or $\inf A$.

Back

Section titled “Back”

For an ordered set $(X, \le)$ and $A \subset X$:

  • Upper Bounds: $A^+ := {x \in X : \forall a \in A, a \le x}$
  • Lower Bounds: $A^- := {x \in X : \forall a \in A, a \ge x}$
  • Supremum (Least Upper Bound): The least element of $A^+$, denoted by $\bigvee A$ or $\sup A$.
  • Infimum (Greatest Lower Bound): The greatest element of $A^-$, denoted by $\bigwedge A$ or $\inf A$.

Lean

Section titled “Lean”

upperBounds / lowerBounds / sSup / sInf