Update EIDMA.mmark

Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>

content/posts/eidma.mmark

@@ -137,3 +137,31 @@ markup = "mmark"
 		- $$(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$$
 
 	- The quotient set: $$\mathbb X / R = \{ [a] : a \in \mathbb X \}$$
+
+## Posets
+
+- Partial orders
+	- $$\mathbb X$$ is a set, $$R \subseteq \mathbb X \times \mathbb X$$
+	- Def: $$R$$ is a partial order on $$\mathbb X$$ iff $$R$$ is:
+		- Reflexive
+		- Antisymmetric
+		- Transitive
+
+	- Def: $$m \in \mathbb X$$ is said to be:
+		- maximal element in $$(\mathbb X, \preccurlyeq)$$ iff $$(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a$$
+		- largest iff $$(\forall a \in \mathbb X) (a \preccurlyeq m)$$
+		- minimal iff $$(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)$$
+		- smallest iff $$(\forall a \in \mathbb X) (m \preccurlyeq a)$$
+
+	- Def: A partial order $$R$$ on $$\mathbb X$$ is said to be *"total"* iff $$(\forall x, y \in \mathbb X) (x R y \lor y R x)$$
+	- Def: A subset $$B$$ of $$\mathbb X$$ is called a chain *"chain"* iff $$B$$ is totally ordered by $$R$$
+    	- $$C(\mathbb X)$$ - the set of all chains in $$(\mathbb X, R)$$
+    	- A chain $$D$$ in $$(\mathbb X, R)$$ is called a maximal chain iff $$D$$ is a maximal element in $$(C(\mathbb X), R)$$
+    	- $$K \subseteq \mathbb X$$ is called an antichain in $$(\mathbb X, R)$$ iff $$(\forall p, q \in K) (p R q \lor q R p \implies p = q)$$
+    	- Def: $$R$$ is a partial order on $$\mathbb X$$, $$R$$ is called a *well* order iff $$R$$ is a total order on $$X$$ and every nonempty subset $$A$$ of $$\mathbb X$$ has the smallest element
+
+## Induction
+- If $$\phi$$ is a propositional function defined on $$\mathbb N$$, if:
+   	- $$\phi(1)$$
+   	- $$(\forall n \geq 1) (\phi(n) \implies \phi(n+1)$$
+	- $$(\forall k \geq 1) \phi(k)$$