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Set theory

set-theory

This article is about the introduction to set theory. In fact, we will give the basic definitions, properties, and universal notions of sets. Part of this course can be used by beginners in algebra. However, part of this article uses advanced mathematics at the college or university level. We notice that such a theory is very important in mathematics, especially in probability and groups theory. So let’s learn this fundamental set theory together.

This concise course is formed by an introduction to set theory and an advanced tool in such theory.

Set theory symbols

set-theory-definition

Definition

In mathematics, a set is just a collection of objects. An object of the set of called an element of the set.

An example of a set is the players of a football team. So the elements of the set are the players.

Take also the example of the real number set it contains all rational and irrational numbers.

It should be noted that a set can contain any type of element, and sometimes a set can contain non-homogeneous elements, such as the set teacher and students. Here we have a single set that contains two types of items, students and teachers.

If $x_1,x_2,x_n,\cdots$ are the elements of a set $S,$ then we use the following definition $$ S=\{x_1,x_2,\cdots\}$$ to denote the set. To say that $x_i$ is an element of $A,$ just write $x_i\in A$. We also say that $x_i$ belongs to the set $A$. For example $\sqrt{2}\in \mathbb{R}$.

To say that an object $a$ is not an element of the set $A,$ we write $a\notin A$. For example $\sqrt{2}\notin \mathbb{Q},$ the set of all rational numbers.

Remark: Imagine a set that contains no elements. It seems strange, yet this set exists in mathematics, and it is called the empty set and will be denoted $\emptyset$.

We say that a set $E$ contains a set $F$, or $F$ is a subset of $E,$ and we denote $F\subset E$ if all elements of $F$ are also elements of $E$. That is for any $x\in F,$ we have $x\in E$. We note that $E$ is a subset of itself, $E\subset E$.

Observe that $A$ and $B$ are subsets of $A\cup B$. Moreover, the intersection $A\cap B$ is a subset of $A$ and $B$.

By convention for any set $A$, we have $\emptyset \subset A$.

If $E$ is a set, we denote by $\mathscr{P}(E)$ the set of all subsets of $E$. Thus we have $$\mathscr{P}(E)=\{A: A\subset E\}.$$

From the sets $A$ and $B$, we can produce other sets as follows

  • The union of sets $A$ and $B$ is a set denoted by $A\cup B$ which contains the elements of $A$ and $B$. Then $x\in A\cup B$ means that $x\in A$ or $x\in B$. As an example, if $A=\{-3,-1,0,\frac{1}{2},\frac{3}{2},7\}$ and $B=\{-\frac{-1}{2},0,2,\frac{5}{2},7\}$, then $$A\subset B=\left\{-3,-1,-\frac{-1}{2},0,\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},7\right\}.$$
  • The intersection of the sets $A$ and $B$ is a set noted $A\cap B$ which contains only the elements belonging to both $A$ and $B$. So that $x\in A\cap B$ means that $x\in A$ and $x\in B$. As an example, take $A=\{\sqrt{2},\frac{1}{2},2\}$ and $B=\mathbb{Q}$. Then $A\cap B=\{\frac{1}{2},2\}$.
  • The product of the sets $A$ and $B$ is a set denoted by $A\times B$ which contains couples $(a,b)$ with $a\in A$ and $b\in B$. So that $$ A\times B:=\left\{(a,b): a\in A,\;b\in B\right\}.$$ As an example the plane $\mathbb{R}\times \mathbb{R}$.

Assume that $A$ is a subset of a set $E$. We define a subset of $E$ by $$ A^c:=\{ x\in E: x\notin A\}.$$ The set $A^c$ is called the complement set of $A$. It is not difficult to see that $E=A\cup A^c$. For example, the set of irrational numbers is exactly the complement set of $\mathbb{Q}$.

If $A$ and $B$ are to set, we define a set called the relative complement of $B$, by $$A\setminus B:=\{x\in A:x\notin B\}.$$ The symmetric difference between the set $A$ and $B$ is the following set $$ A\Delta B=(A\setminus B)\cup(B\setminus A).$$ Clearly, we have $$ A\Delta B=\{ x\in A\subset B:x\notin A\cap B\}.$$

The cardinality of a set $A$ is the number of elements of $A$. Generally, it is denoted by $|A|,$ but also in some algebra books by $n(A),$ ${\rm card}(A),$ or #$ A$. As example, for $A=\{-1,6,10,17\},$ we have $|A|=4$.

Operations on sets

Assume that we have a set $M$ that satisfies the following property: there exists an application on $M\times M$ denoted by $\ast$ such $(x,y)\in M\times M\mapsto x\ast y\in M$. This application is called a composition law on $M$. Now let $A$ and $B$ two subsets of $M$. We can then define a subset of $M$ by $$A\ast B:=\{ a\ast b: a\in A,\;b\in B\}.$$ Let us give some examples. If we take $M=\mathbb{R}$ and $\ast=+,$ the addition operation in $\mathbb{R},$ then we can define the sum of two subset $A$ and $B$ on $\mathbb{R}$ by $x\in A+B$ if and only if $x=a+b$ with $a\in A$ and $b\in B$. Moreover, if we take $\ast=\cdot$, the standard multiplication operation on $\mathbb{R}$, we can define a set $A\cdot B$ by $x\in A\cdot B$ if and only if $x=a\cdot b$ for $a\in A$ and $b\in B$.

We say that two sets $A$ and $B$ are equal and we write $A=B$ if and only if $A\subset B$ and $B\subset A$.

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