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R-polar

  • R-polar #Card
    • For a relation $R \subset X \times Y$ and $A \subset X$, the $R$-polar of $A$ is defined as: $$R^\circ A := \bigcap_{x \in A} Rx \subset Y$$ Properties: - $A_0 \subset A_1 \implies R^\circ A_0 \supset R^\circ A_1$ - $A \subset (R^-)^\circ (R^\circ A)$

Back

Section titled “Back”

For a relation $R \subset X \times Y$ and $A \subset X$, the $R$-polar of $A$ is defined as: $$R^\circ A := \bigcap_{x \in A} Rx \subset Y$$ Properties:

  • $A_0 \subset A_1 \implies R^\circ A_0 \supset R^\circ A_1$
  • $A \subset (R^-)^\circ (R^\circ A)$

Lean

Section titled “Lean”

本ノート RoyalRoad.ChapterI.polar R A := {y | ∀ x ∈ A, (x,y) ∈ R}