Question

Write $\left( -7,5 \right),\left[ -6,12 \right],\left( 3,7 \right],\left[ -16,6 \right)$ in set builder form.

Answer 1

Hint: We have been given various sets which are continuous intervals of various numbers. We shall write these intervals of sets in set builder form by using the required mathematical symbols and assuming a variable in each set.

Complete step by step solution:
In order to write a set in set builder form, we first write a capital letter to denote the name of the set, then put an equal sign and open the left curly brace. We pick a letter as a variable (very often the letter is a lower-case x but it could be any letter we want).
After that we put a bar and describe what x could be. Since mathematics is more universal than just a single language, mathematicians rather write in symbols.
$\left( -7,5 \right)$ in set builder form is given as $\left\{ x:x\in \mathbb{R}\text{ and }-7 < x < 5 \right\}$ where $\mathbb{R}$ denotes real numbers because the interval does not include -7 and 5.
$\left[ -6,12 \right]$ in set builder form is given as $\left\{ x:x\in \mathbb{R}\text{ and }-6\le x\le 12 \right\}$ because -6 and 12 are included in the set interval.
$\left( 3,7 \right]$ in set builder form is given as $\left\{ x:x\in \mathbb{R}\text{ and 3} < x\le 7 \right\}$ because 3 is not included but 7 is included in the interval.
$\left[ -16,6 \right)$ in set builder form is given as $\left\{ x:x\in \mathbb{R}\text{ and }-16\le x < 6 \right\}$ because -16 is included but 6 is not included in the interval.
Therefore, $\left( -7,5 \right)$ in set builder form is $\left\{ x:x\in \mathbb{R}\text{ and }-7 < x < 5 \right\}$, $\left[ -6,12 \right]$ in set builder form is given as $\left\{ x:x\in \mathbb{R}\text{ and }-6\le x\le 12 \right\}$, $\left( 3,7 \right]$ in set builder form is given as $\left\{ x:x\in \mathbb{R}\text{ and 3} < x\le 7 \right\}$ and $\left[ -16,6 \right)$ in set builder form is given as $\left\{ x:x\in \mathbb{R}\text{ and }-16\le x < 6 \right\}$.

Note:
There are two ways to write a set. One way is just to list all the elements of the set inside curly braces separated by commas. This way is known as the roster method of set representation. The other way is the set builder form in which the set is defined with respect to some variable and the interval in which its components lie are given by using greater than or less than sign with the predefined variable letter.