diff --git a/content/posts/eidma.mmark b/content/posts/eidma.mmark
index d80aaae..bcef92b 100644
--- a/content/posts/eidma.mmark
+++ b/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: $$f^{-1}(\mathbb A)$$
diff --git a/public/index.html b/public/index.html
index 16eba8c..a78eb79 100644
--- a/public/index.html
+++ b/public/index.html
@@ -141,10 +141,9 @@
Introduction to Discrete Mathematics
\((\forall n \geq 1) (\phi(n) \implies \phi(n+1)\)
+
\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)
\((\forall k \geq 1) \phi(k)\)
+
+
+
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: \(f^{-1}(\mathbb A)\)
diff --git a/public/posts/index.html b/public/posts/index.html
index 5a604c1..1e0b5b0 100644
--- a/public/posts/index.html
+++ b/public/posts/index.html
@@ -138,10 +138,9 @@
Introduction to Discrete Mathematics