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This is a well-known joke, so many of you might already be familiar with it. Nonetheless, it’s quite amusing as a joke. 😆 The Assertion: “Everyone is Bald” As indicated in the title, we will prove the assertion that “everyone is bald” using mathematical induction (often simply called induction). Understanding Mathematical Induction Let \( P(n)…

Definition Let \( K \) be a subset of the \( n \)-dimensional Euclidean space \( \mathbb{R}^n \). When any line segment \( \overline{xy} \) that connects two any points \( x \) and \( y \) in \( K \) is also contained within \( K \), we say that \( K \) is…

Laplacian The Laplacian on \(\mathbb{R}^3 \), \[ \Delta = \left( \frac{\partial^2}{\partial x^2},\ \frac{\partial^2}{\partial y^2},\ \frac{\partial^2}{\partial z^2 } \right), \] is a second-order differential operator. When this operator is expressed in spherical polar coordinates in \( \mathbb{R}^3 \), where \[ x = r \sin \theta \cos \varphi,\ y = r \sin \theta \sin\varphi,\ z = r…

The following theorem is called the Jordan Curve Theorem. Definition Let \( c \) be a loop⭕️ in the plane (also known as a Jordan curve or simple closed curve). The complement of \( c \) in \( \mathbb{R}^2 \), denoted \( \mathbb{R}^2-c \), consists of a bounded domain (the interior) and an unbounded domain…

Simply connected is a fundamental concept in mathematics, particularly in topology. Topology studies how the shape of a space behaves and how it can be continuously deformed. A space is called simply connected if any loop (closed curve) within it can be continuously shrunk to a single point. A space with this property is referred…