diff --git a/content/posts/eidma.mmark b/content/posts/eidma.mmark index 809b8b0..d942b55 100644 --- a/content/posts/eidma.mmark +++ b/content/posts/eidma.mmark @@ -115,10 +115,10 @@ markup = "mmark" - Def: A relation $$R$$ on a set $$\mathbb X$$ is a subset of $$\mathbb X \times \mathbb X$$ ($$R \subseteq \mathbb X \times \mathbb X$$) - Graph of a function $$f()$$: $$\{ (x, f(x) : x \in Dom(f) \}$$ - Properties of: - 1. Reflexivity: $$(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$$ - 2. Symmetricity: $$[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$$ - 3. Transitivity: $$(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$$ - 4. Antisymmetricity: $$(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$$ + - Reflexivity: $$(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$$ + - Symmetricity: $$[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$$ + - Transitivity: $$(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$$ + - Antisymmetricity: $$(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$$ - Equivalence relations: - Def: $$R \subseteq \mathbb X \times \mathbb X$$ is said to be an equivalence relation *iff* $$R$$ is reflexive, symmetric and transitive. diff --git a/public/posts/eidma/index.html b/public/posts/eidma/index.html index d62cf2c..515b996 100644 --- a/public/posts/eidma/index.html +++ b/public/posts/eidma/index.html @@ -345,12 +345,12 @@

  • Properties of:

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  • Equivalence relations: