Mathematical Proof/Introduction to Set Theory

Objects known as sets are often used in mathematics, and there exists which studies them. Although set theory can be discussed formally, it is not necessary for us to have such a formal discussion in this book, and we may not be interested in and understand the formal discussion in this stage.

Even if we do not discuss set theory formally, it is important for us to understand some basic concepts about sets, which will be covered in this chapter.

What is a set?
A may be viewed as a collection of well-defined, distinct objects (the objects can also be sets). Because of the vagueness of the term "well-defined", we do not regard this as the definition of set. Instead, we regard set as a primitive notion (i.e., concepts not defined in terms of previously-defined concepts). Other examples of primitive notions in mathematics include and.

We have mentioned that a set is a collection of well-defined, distinct objects. Objects in a set are called of the set. We write $$a\in A$$ to mean that the element $$a$$ belongs to the set $$A$$. If $$a$$ does not belong to $$A$$, we write $$a\notin A$$.

Ways of describing a set
There are multiple ways to describe a set precisely (in the sense that element(s) belonging to the set is (are) known precisely).

If a set consists of a small number of elements, then the may be quite efficient. In the listing method, elements of a set are listed within a pair of braces ({}). In particular, just changing the listing of elements does change the set represented. For example, $$\{1,2\}$$ and $$\{2,1\}$$ are both representing the same set whose elements are 1 and 2. If the elements listed in the pair of braces are the same, the notations created by the listing method with different listing orders refer to the set. Also, repeatedly listing a specific element in a set does change the set represented. For example, $$\{1,1,2,2,2\}$$ and $$\{1,2\}$$ are both representing the same set whose elements are 1 and 2. In particular, if a set contains no elements, it can be denoted by $$\{\}$$ based on the listing method or $$\varnothing$$. This kind of set is called an empty set.

Another way to describe a set is using. For example, consider the set $$S$$ of prime numbers less than 10. If we use the listing method instead, the set $$S$$ is represented by $$\{2,3,5,7\}$$.

The third way to describe a set is advantageous when a set contains many elements. This method is called. There are parts within a pair of braces in this notation. They are illustrated below with descriptions: $$\underbrace{\{}_{\text{The set of}}\underbrace{x}_{\text{all elements }x}\underbrace{:}_{\text{such that}}\underbrace{P(x)}_{\text{the property }P(x)\text{ is satisfied}}\}$$

As one may expect, two sets are if and only if they contain the same elements. Equivalently, two sets $$A$$ and $$B$$ are equal if and only if each element of $$A$$ is also an element of $$B$$ and each element of $$B$$ is also an element of $$A$$. This can be regarded as an axiom or a definition. If two sets $$A$$ and $$B$$ are equal, we write $$A=B$$. If not, we write $$A\ne B$$.

In this book, when we are solving an equation, we are only considering its solution(s) unless stated otherwise.

{{colored exercise| Assume $$a$$ and $$b$$ are different elements. Is each of the following statements true or false? {The set $$\big\{\{\}\big\}$$ contains no elements. - True. + False.
 * type=""}
 * This set contains one element, namely the empty set $$\{\}$$.

{$$\{\}=\varnothing.$$ + True. - False.
 * type=""}
 * These two sets have no elements, thus containing the "same" elements.

{$$\{1,1,2,3\}$$ is a set. + True. - False.
 * type=""}
 * Even if the element "1" is listed two times, it still represents a set (which can also be denoted by $$\{1,2,3\}$$).

{$$\{x:0<x<2\text{ and }x^2=4\}=\{2\}$$. - True. + False. }
 * type=""}
 * Solving $$x^2=4$$, we get $$x=\pm 2$$, but both of them are not between 0 and 2 exclusively, so no $$x$$ satisfies the given property. As a result, the set on the RHS should be empty.

{$$\{a,b,b\}\in\{\{a,b\},a,b,b\}$$. + True. - False.
 * type=""}
 * First, $$\{a,b,b\}=\{a,b\}$$. Since the set $$\{a,b\}$$ is an element of the set on RHS, this is true.

{$$\{\{a\},\{a\}\}\in\{\varnothing,\{a\},\{b\},\{a,b\}\}$$. - True. + False. }}
 * type=""}
 * First, $$\{\{a\},\{a\}\}=\{\{a\}\}$$. But set $$\{\varnothing,\{a\},\{b\},\{a,b\}\}$$ doesn't contains $$\{\{a\}\}$$, it only contains $$\{a\}$$, which is not the same.

Set cardinality
If a set contains number of elements, it is called a  set, and it is called an  set otherwise. If a set is finite, then its cardinality is its number of elements. For infinite sets, it is harder and more complicated to define their cardinalities, and so we will do this in the later chapter about set cardinalities. For each set $$S$$, its cardinality is denoted by $$|S|$$.

There are some special infinite sets for which notations are given, as follows: In particular, we can use set-builder notation to express $$\mathbb Q$$, as follows: $$\mathbb Q=\{p/q:p,q\in\mathbb Z\text{ and }q\ne 0\}$$.
 * $$\mathbb N$$ is the set of all natural numbers (0 is regarded as a natural number in this book).
 * $$\mathbb Z$$ is the set of all integers.
 * $$\mathbb Q$$ is the set of all rational numbers.
 * (nonstandard notation) $$\mathbb I$$ is the set of all irrational numbers.
 * $$\mathbb R$$ is the set of all real numbers.
 * $$\mathbb C$$ is the set of all complex numbers.

Subsets
We call some commonly encountered subsets of $$\mathbb R$$. For each real number $$a,b$$ such that $$a<b$$, There are also some intervals: Note: $$\{x\in S:P(x)\}$$ is a shorthand of $$\{x:x\in S\text{ and }P(x)\}$$ ($$S$$ is a set).
 * $$(a,b)\overset{\text{ def }}=\{x\in\mathbb R:a<x<b\}$$ (open intervals)
 * $$(a,b]\overset{\text{ def }}=\{x\in\mathbb R:a<x\le b\}$$ (half-open (or half-closed) intervals)
 * $$[a,b)\overset{\text{ def }}=\{x\in\mathbb R:a\le x< b\}$$ (half-open (or half-closed) intervals)
 * $$[a,b]\overset{\text{ def }}=\{x\in\mathbb R:a\le x\le b\}$$ (closed intervals)
 * $$(-\infty,a)\overset{\text{ def }}=\{x\in\mathbb R:x<a\}$$
 * $$(-\infty,a]\overset{\text{ def }}=\{x\in\mathbb R:x\le a\}$$
 * $$(a,\infty)\overset{\text{ def }}=\{x\in\mathbb R:a<x\}$$
 * $$[a,\infty)\overset{\text{ def }}=\{x\in\mathbb R:a\le x\}$$
 * $$(-\infty,\infty)\overset{\text{ def }}=\mathbb R$$

Universal set and Venn diagram
A is a diagram showing all possible logical relationships between a finite number of sets. The universal set is usually represented by a region enclosed by a rectangle, while the sets are usually represents by regions enclosed by circles. The following is a Venn diagram. In this diagram, if the white region represents set $$A$$ and the region enclosed by the rectangle represents the universal set, then the red region is the set $$A^c$$.

However, the following is a Venn diagram. This is because there are not regions in which only the yellow and blue region intersect, and only the red and green region intersect, respectively. So, not logical relationships between the sets are shown.

To show all logical relationships between four sets, the following Venn diagram can be used.

Set operations
Similar to the arithmetic operations for real numbers which combine two numbers into one, the set operations combine two sets into one.

Cartesian product
Similarly, we can define the Cartesian product of three or more sets.