Set Theory
8.0 Intervals as subsets of $R$
8.0 Intervals as subsets of $R$
- Closed interval
Let $a$ and $b$ be two given real numbers such that $a < b$, then the set of all real numbers $x$ such that $a \le x \le b$ is called a closed interval and is denoted by $$[a,b]$$
$$[a,b] = \{ x \in R:a \le x \le b\} $$
The completely shaded dots represent the closed interval.
- Open interval
If $a$ and $b$ are two real numbers such that $a < b$, then the set of all real numbers $x$ such that $a < x < b$ is called an open interval and is denoted by $$(a,b)$$
$$(a,b) = \{ x \in R:a < x < b\} $$
The hollow shaded dots represent the open interval.
- Semi-open or Semi-closed interval
If $a$ and $b$ are two real numbers such that $a < b$, then the sets
$$\begin{equation} \begin{aligned} (a,b] = \{ x \in R:a < x \le b\} \\ [a,b) = \{ x \in R:a \le x < b\} \\\end{aligned} \end{equation} $$
are called an semi-open interval or semi-closed intervals.
Illustration 16. Classify each set according to the intervals.
a. $A = \{ x:x \in R, - 5 \le x < 10\} $
b. $B = \{ x:x \in R,10 < x \le 23.2\} $
c. $C = \{ x:x \in R, - 1 \le x \le 1\} $
d. $D = \{ x:x \in R,0 < x < 5.2\} $
e. $E = \{ x:x \in R, - \infty < x < \infty \} $
f. $F = \{ x:x \in R, - 2.5 \le x < \infty \} $
Solution:
| SET | CLASSIFICATION |
| a. $A = \{ x:x \in R, - 5 \le x < 10\} $ | This can be written as, $[ - 5,10)$. The set $A$ is open ended on one side. Hence, this is a semi-open or semi-closed interval. |
| b. $B = \{ x:x \in R,10 < x \le 23.2\} $ | This can be written as, $(10,23.2]$ . The set $B$ is open ended on one side. Hence, this is a semi-open or semi-closed interval. |
| c. $C = \{ x:x \in R, - 1 \le x \le 1\} $ | This can be written as, $[ - 1,1]$ . The set $C$ is closed interval. |
| d. $D = \{ x:x \in R,0 < x < 5.2\} $ | This can be written as, $(0,5.2)$ . The set $D$ is an open interval. |
| e. $E = \{ x:x \in R, - \infty < x < \infty \} $ | This can be written as, $( - \infty ,\infty )$. The set $E$ is an open interval. |
| f. $F = \{ x:x \in R, - 2.5 \le x < \infty \} $ | This can be written as, $[ - 2.5,\infty )$. The set $F$ is open ended on one side. Hence, this is a semi-open or semi-closed interval. |
Illustration 17. Describe the interval for the following sets.
a. Set of numbers that are less than the even prime and greater than $-5$
b. Set of all non-negative numbers
c. $C = \{ x:x \in R, - 1 \le x \le 1\} $
d. Set of numbers less than or equal to the first two digit natural number and greater than $-12$
e. Set of all numbers less than the first natural composite number
f. Set of real numbers
Solution:
| SET | INTERVAL |
| a. Set of numbers that are less than the even prime and greater than $-5$ | The interval of the set is $( - 5,2)$ |
| b. Set of all non-negative numbers | The interval of the set is $[0,\infty )$ |
| c. $C = \{ x:x \in R, - 1 \le x \le 1\} $ | The interval of the set is $[ - 1,1]$ |
| d. Set of numbers less than or equal to the first two digit natural number and greater than $-12$ | The interval of the set is $( - 12,10]$ |
| e. Set of all numbers less than the first natural composite number | The interval of the set is $( - \infty ,4)$ |
| f. Set of real numbers | The interval of the set is $( - \infty ,\infty )$ |
Illustration 18. Write the following intervals in set builder form.
a. $[ - 3,8)$
b. $[ - 2,5]$
c. $(3,52)$
d. $(12,23]$
e. $( - \infty ,\infty )$
Solution:
| SET | SET-BUILDER FORM |
| a. $[ - 3,8)$ | $\{ x: - 3 \le x < 8,x \in R\} $ |
| b. $[ - 2,5]$ | $\{ x: - 2 \le x \le 5,x \in R\} $ |
| c. $(3,52)$ | $\{ x:3 < x < 52,x \in R\} $ |
| d. $(12,23]$ | $\{ x:12 < x \le 23,x \in R\} $ |
| e. $( - \infty ,\infty )$ | $\{ x:x \in R\} $ |
