Real Numbers

We review the numbers which will be used in this lecture. Natural numbers are the numbers used for counting such as $1,2,3,4,\ldots$. Integers are the natural numbers with 0 and negative numbers such as $\ldots,-3,-2,-1,0,1,2,3,\ldots$. Rational numbers are fractions of integers $\frac{m}{n}$ with non-zero denominator $n$. Irrational number are numbers with non repeated infinite decimals such as $\sqrt{2}, \sqrt{3}, \pi$. Real numbers are either rational numbers or irrational numbers. Furthermore, $i$ satisfying $i^2 = -1$ is called the imaginary unit. The set of real numbers can be expressed either using the symbol ${\mathbb{R}}$ ^0.1The real numbers can be thought of as points on a number line. In other words, every real number can be put into one-on-one correspondence with the point on the number line.

$$\begin{array}{lllllllll}\hline \hspace{1cm} & -3& -2 & -1 & 0 & 1 & 2 & 3 & \hspace{1cm}\\ \end{array}$$

For $x \in {\mathbb{R}}$, we define

$$\vert x\vert = \left\{\begin{array}{ll} x & (x \geq 0)\\ -x & (x < 0) \end{array}\right.$$

We call $\vert x\vert$ absolute value of $x$. For example, $\vert 3\vert = 3, \vert-4\vert = 4$. Now carefully look at $\sqrt{x^2}$. For if $x \geq 0$, then we have $\sqrt{x^2} = x$. For if $x < 0$, then we have $\sqrt{x^2} = -x$. This means that $\sqrt{x^2}$ and $\vert x\vert$ give rise the same number. Thus, we can say $\sqrt{x^2} = \vert x\vert$. For example, $\sqrt{3^2} = 3, \sqrt{(-4)^2} = -(-4) = 4$.

Element of the set

A distinct object $a$ belongs to the set $A$ is called the element of the set $A$ and denoted by $a \in A$.

NOTE The set of natural numbers is ${\mathbb{N}}$. Then we write $3 \in {\mathbb{N}}$ and say 3 is an element of ${\mathbb{N}}$. 0 is not a natural number. Then we write $0 \not\in {\mathbb{N}}$.

Subset

If all elements of $B$ are also elments of $A$, then $B$ is subset of $A$ and denoted by $B \subset A$.

NOTE For ${\mathbb{N}}, {\mathbb{Q}}, {\mathbb{R}}$, we must have ${\mathbb{N}} \subset {\mathbb{Q}} \subset {\mathbb{R}}$.

Union

The Union of tow sets $A$ and $B$ is the collection of points which are in $A$ or in $B$ or in $A$ and $B$, and denoted by $A \cup B$.

NOTE The set $A \cup B$ consists of all elements of $A$ and elements of $B$.

Intersection

The Intersection of two sets $A$ and $B$ is the collection of points which are in $A$ and $B$ and denoted by $A \cap B$.

NOTE The set $A \cap B$ consists of elements which have both properties of $A$ and $B$.

Subsets of ${\mathbb{R}}$

$$\begin{array}{lll} & & \begin{array}{lllll} \hspace{1cm} &\h... ...cm} \bullet &\hspace{-0.3cm} \hspace{1cm}\end{array}\end{array}$$