Quartz sync: Mar 1, 2025, 2:26 PM

content/Definitions/Functions/Characteristic Function.md

@@ -0,0 +1,4 @@
+$\varkappa_{y} \in \Pi_{X} \, \{ 0, \, 1 \}$, so $\varkappa_{y} : X \to \{ 0, \, 1 \}$ defined as:
+$$\varkappa_{y}(x) = \begin{cases}
+1, & \text{if}\ x \in Y\\ 0, & \text{if}\ x \notin Y
+\end{cases}$$

content/Definitions/Functions/Direct Product.md

@@ -0,0 +1,2 @@
+$$X \times Y = \{ x: \{ 1, 2 \} \to x_{1} \cup x_{2} | x_{1} X_{1} \wedge x_{2} \in X_{2} \}$$
+where $X = X_{1}$ and $Y = X_{2}$, basically $(X_{1}, X_{2})$.

content/Definitions/Functions/Inverse Function.md

@@ -0,0 +1,4 @@
+$$f^{-1} : Y \to X$$
+such that
+$$ff^{-1} = I_{x} \; \land f^{-1}f = I_{y}$$
+$$\exists f^{-1} \iff f \; \text{bijective}$$

content/Definitions/Functions/Metric.md

@@ -0,0 +1,2 @@
+# Definition
+A function which measures distances between two points in a [[Metric Space]].

content/Definitions/Functions/Power Set.md

@@ -0,0 +1,10 @@
+# Definition
+$\wp(X)$ of $X$ consists of all the subsets of $X$.
+
+## Amount of Elements in a Power Set
+Lets say we have $|X|$:
+$$|X| = |\{ 1, \, \dots, \, n \}|$$
+The $\wp(X)$ would have $2^{n}$ elements in the set.
+# Example
+$$X = \{ 1, \, 2 \}$$
+$$\wp(X) = \{ \emptyset, \, \{ 1 \}, \, \{ 2 \}, \, X \}$$