Update eidma

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

content/posts/eidma.mmark

@@ -1,10 +1,7 @@
 +++
 title = "Introduction to Discrete Mathematics"
-date = "2019-11-04"
-author = "Abdulkadir"
-showFullContent = false
+date = "2019-11-20"
 tags = ["university-notes"]
-markup = "mmark"
 +++
 
 - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
@@ -161,7 +158,23 @@ markup = "mmark"
     	- 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:
    	- $$\phi(1)$$
-   	- $$(\forall n \geq 1) (\phi(n) \implies \phi(n+1)$$
+   	- $$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$$
 	- $$(\forall k \geq 1) \phi(k)$$
+
+## Functions
+
+- $$f: \mathbb X \to \mathbb Y$$
+- Def: $$f \subseteq \mathbb X \times \mathbb Y$$ is said to be a function if:
+  - $$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$$
+  - $$(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$$
+
+- Types of functions $$f: \mathbb X \to \mathbb Y$$:
+  - $$f$$ is said to be an injection ( 1 to 1 function) iff $$(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$$
+  - $$f$$ is said to be a surjection (onto function) iff $$(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$$
+  - If $$f^{-1}$$ is a function from $$\mathbb Y \to \mathbb X$$ then $$f^{-1}$$ is called the inverse function for $$f$$
+      - Fact: $$f^{-1}$$ is a function iff $$f$$ is a *bijection* (1 to 1 and onto)
+
+- For some set $$\mathbb A$$ the image of $$\mathbb A$$ by $$f$$ is $$f(\mathbb A) = \{ f(x) : x \in \mathbb A \}$$. We can also define the inverse of an image even when the function itself isn't invertible: $$f^{-1}(\mathbb A)$$