Inequality and intervals

Inequality between two numbers

We use the inequality symbols $<, \leq, >$ and $\geq$ to compare the size of two real numbers. For example, the following statements are correct:

$1<2$ : say " $1$ is less than $2$ ".
$1\leq 2$ : say " $1$ is less than or equal to $2$ " Note that the statement $2\leq 2$ is correct, but the strict version $2<2$ is not.
$2>1$ : say " $2$ is greater than $1$ ".
$2\geq 1$ : say " $2$ is greater than or equal to $1$ ". Again, the statement $2\geq 2$ is correct, but the stricter version $2>2$ is not.

We can also use inequality symbols to define subsets of the number sets. Here are some examples:

Example 1

$A$ is the set of all natural numbers $x$ with $x\leq 7$ . Using a more formal notation, we could write
$A=\{ x\in \mathbb{N}\,|\, x\leq 7\}$
It is $A=\{1,2,3,4,5,6,7\}$
$B$ is the set of all natural numbers $x$ with $x< 7$ . Using a more formal notation, we could write
$B=\{ x\in \mathbb{N}\,|\,x< 7\}$
It is $B=\{1,2,3,4,5,6\}$
$C$ is the set of all integers $x$ with $x>-3.5$ . Using a more formal notation, we would write
$C=\{ x\in \mathbb{Z}\,|\,x>-3.5\}$
It is $C=\{-3,-2,-1,0,1,...\}$
$D$ is the set of all natural numbers $x$ with $1\leq x <6$ . Note that with this notation we mean that $1\leq x$ and $x<6$ (that is, $x$ is between $1$ and $6$ , where $1$ is in the set, but $6$ is not). Using a more formal notation, we write
$D=\{ x\in \mathbb{N}\,|\,1\leq x <6\}$
It is $D=\{1,2,3,4,5\}$

Of course we can replace $\mathbb{N}$ with any other number set such as $\mathbb{Z}, \mathbb{Q}, \mathbb{R}$ .

Exercise 1

Determine the elements of set $A$ :

$A$ is the set of all natural numbers $x$ with $x>6$ .
$A=\{x \in \mathbb{Z}\,|\, -4.75\leq x\leq 2\}$
$A$ is the set of all rational numbers $\frac{p}{q}$ with $1<p<4$ and $12\leq q\leq 14$ .
$A=\{x \in \mathbb{N}\,|\, 4x<28\}$

Solution

$A=\{7,8,9,...\}$
$A=\{-4,-3,-2,-1,0,1,2\}$
$A=\{\frac{2}{12},\frac{2}{13},\frac{2}{14},\frac{3}{12},\frac{3}{13},\frac{3}{14}\}$
$A=\{1,2,3,4,5,6\}$

Intervals

Intervals are segments (or subsets) of the number line (the real numbers). For example, all numbers between $2$ and $5$ form an interval. Emphasis is on all numbers, so not just the natural numbers between $2$ to $5$ , or just the fractions between $2$ and $5$ , but also the irrational numbers such as $\pi=3.14...$ or $\sqrt{5}=2.236...$ .

We use square brackets to indicate intervals, and distinguish between intervals which do or do not include the borders:

$[1,2]$ The set of all real numbers between $1$ and $2$ , including the borders $1$ and $2$ . Thus, we have
$[1,2] = \{x\in \mathbb{R}\,|\, 1\leq x\leq 2\}$
$[1,2[$ The set of all real numbers between $1$ and $2$ , including the left border $1$ but without the right border $2$ ). Thus, we have
$[1,2[ = \{x\in \mathbb{R}\,|\, 1\leq x< 2\}$
$]1,2]$ The set of all real numbers between $1$ and $2$ , without the left border $1$ but including the right border $2$ ). Thus, we have
$]1,2] = \{x\in \mathbb{R}\,|\, 1< x\leq 2\}$
$]1,2[$ The set of all real numbers between $1$ and $2$ , without the left and right borders). Thus, we have
$]1,2[ = \{x\in \mathbb{R}\,|\, 1< x< 2\}$
More generally, if $a$ and $b$ denote two real numbers, where $b$ is greater than or equal to $a$ , we can define the intervals $[a,b], [a,b[, ]a,b]$ and $]a,b[$ . They all denote the set of all real numbers between $a$ and $b$ , but depending on how we write the square brackets, the borders $a$ and $b$ are included or not.

Note that the borders can also be an infinitely big number (which we denote by $-\infty$ and $\infty$ . We then have
$[1,\infty[$ The set of all real numbers greater than or equal to $1$ :
$[1,\infty[=\{x\in \mathbb{R}\,|\, x\geq 1\}$
$]1,\infty[$ The set of all real numbers greater than $1$ :
$]1,\infty[=\{x\in \mathbb{R}\,|\, x>1\}$
$]-\infty,2]$ The set of all real numbers less than or equal to $2$ :

]-\infty,2]=\{x\in \mathbb{R}\,|\, x\leq 2\}

$]-\infty,2[$ The set of all real numbers less than $2$ :
$[-\infty,2[=\{x\in \mathbb{R}\,|\, x<2\}$
$]-\infty,\infty[$ The real numbers $\mathbb{R}$

$-\infty$ and $\infty$ are borders which are never included, as they are not real numbers.

Exercise 2

Express the segments drawn below on the number line by intervals. Several intervals may have to be linked by set operations. The dotted segments in 3 and 4 means that they go on and on.

Solution

$[2,4[$
$[-3.102, 4.1\overline{3}]$
$[1,\infty[$
$]-\infty,2[$
$[1,2]\cup [2.5,4]\cup [4.5,7]$

Exercise 3

Express the following sets using intervals or the union of intervals. Hint: Indicate the sets on the number line. Also note that for intervals the universal set is always $\mathbb{R}$ .

$\{1\}$
$[1,2[^c$
$]-0.5,3[ \,\cap\, [1.5,2[$
$\mathbb{R}$
$\{x\in \mathbb{R}\,|\, 2x-1>2\}$
$]1,1[^c$
$[1,1]^c$
$[1,3]\,\setminus \,]0.5,2[$
$[1,2[\,\cup\, \{2\}$

Solution

$[1,1]$
$]-\infty,1[ \,\cup\, [2,\infty[$
$[1.5,2[$
$]-\infty,\infty[$
$]1.5,\infty[$
$\{\}^c=\mathbb{R}$
$\{1\}^c=]-\infty,1[ \,\cup\, ]1,\infty[$
$[2,3]$
$[1,2]$