## Equations of Mathematical Physics, Notes on Set Theory

This is a summary of Chapter 1, Section 1 of Reference [1], which is *Equations of Mathematical Physics* by V.S. Vladimirov. The majority of this post consists of definitions from V.S. Vladimirov.

## Terms in Set Theory

### Foundation

$A$ is a set.

$a \in A$ means element $a$ is contained in set $A$.

$B$ is another set.

$A \subset B$ means $A$ is contained in $B$.

$A \cup B$ means the union of $A$ and $B$.

$A \cap B$ means the intersection of $A$ and $B$.

$A \backslash B$ means the complement of $B$ relative to $A$.

$A \times B$ is called the product of $A$ and $B$ and means the set of pairs $(a,b)$, $a \in A$, $b \in B$.

$\varnothing$ is the symbol for an empty set.

$A=B$ means $A$ is coincident with $B$.

### Points

$R^n$ means $n$-dimensional Euclidean space.

$x = (x_1,x_2,…,x_n)$ is the **set of** points belonging to the set $R^n$.

- Perhaps this $x$ should have an index such as $\gamma$, to keep track of all the different points.

$(x,y) = x_1 y_1 + x_2 y_2 + … + x_n y_n$ is called a scalar product.

$|x| = \sqrt{x_1^2 + x_2^2 + … + x_n^2}$ is called the norm of $x$.

“$|x-y|$ is the Euclidean distance between $x$ and $y$.”

“$|x – x_0| < R$ is called an *open sphere* with radius $R$ and center $x_0$ and denoted by $U(x_0; R)$.” Also, $U_R = U(0;R)$.

- Note that if $R=0$, $|x – x_0| < 0$, but $|x – x_0| \ge 0$ by definition, so $R$ cannot equal zero. $R$ must be positive to satisfy the inequality; $R>0$

### Sequence of Points

“The sequence of points $x_k = (x_{1k}, x_{2k}, x_{3k}, …, x_{nk})$, $k=1,2,…,$ is said to *converge* to the point $x$ in $R^n$, written $x_k \rightarrow x$ as $k \rightarrow \infty$, if $|x_k – x| \rightarrow 0$ as $k \rightarrow \infty$.”

*Note that $k$ is an index. It helps to picture $k$ as a independent variable on a line, with $x_k$ as a dependent variable.*

“The sequence $x_k, k=1,2,…,$ is said to *converge in itself* in $R^n$ if $|x_k – x_p| \rightarrow 0$ as $k \rightarrow \infty$, $p \rightarrow \infty$.”

### Cauchy’s Principle of Convergence

“Cauchy’s Principle of convergence: In order that a sequence of points should converge in $R^n$, it is *necessary* and *sufficient *that it should converge in itself in $R^n$.”

### Necessary and Sufficient Conditions

Reference [2] provides definitions of necessary and sufficient conditions. They are as follows.

“If A, then B” is called a

necessary condition.“A, only if B” is called a

sufficient condition.

### Bounded, Open, Connected, Closed, Compactum

“A set is said to be *bounded in $R^n$* if there is a sphere containing this set.”

“$x_0$ is called an interior point of a set if there is a sphere $U(x_0;R)$ contained in this set.”

“A set is called *open *if all its points are interior.”

“A set is called *connected* if any two of its points can be joined by an unbroken line lying this set.”

- I assume this line can be curved.

“A connected open set is called a *region*.”

“The point $x_0$ is called a limit point of the set $A$ if there is a sequence $x_k, k=1,2,…,$ such that $x_k \in A$, $x_k \rightarrow x_0$ as $k \rightarrow \infty$.”

“If we add all limit points of a set $A$ to the set $A$, the set obtained is called the *closure *of the set A and is denoted by $\bar{A}$; it is clear that $A \subset \bar{A}$.”

“If a set coincides with its closure, it is called *closed*.”

“A closed bounded set is called a *compactum*.”

### Neighborhood of a Set

“Each open set containing $A$ is called a *neighborhood* of the set $A$.”

“The union of the spheres $U(x; \varepsilon)$, when $x$ ranges over $A$, written symbolically as $A_{\in} = \cup_{x \in A} U(x;\varepsilon)$, is called an $\varepsilon$-neighborhood of the set $A$.”

### Characteristic Function of a Set

“The function $\chi_{A}(x)$, which is equal to 1 when $x \in A$ and to 0 when $x \not\in A$, is called the characteristic function of the set $A$.”

### Comment

Regardless of how close two different points are in physical space, these points have coordinates which are numerically different, so there is a nonzero Euclidean distance between these two points. This nonzero distance can be thought of as the radius of an open sphere.

### Boundary of a Region

“Let $G$ be a region.”

- Recall that a region is connected and
**open**. - A set is
**open**if all points are interior–that is, an open sphere can be centered around each point.

“The points of”

“closure $\bar{G}$ not belonging to $G$”

“form a closed set $S$, called the *boundary *of the region $G$, so that

$S = \bar{G} \backslash G$.”

“The connected part of the set $S \cap U(x_0;R)$ which contains the point $x_0$ is called the *neighborhood* of the point $x_0$ on the surface $S$.

- Do not confuse a
*neighborhood of a set*with*the neighborhood of a point on a surface*.