Bounds and Extrema
QUESTION
Bounds and Extrema
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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 $\inf A$.
Click to see question - ソースノート: chapter1.md
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