Question

Write the interval $\left( {6,12} \right)$ in set builder form.

Answer 1

To change an interval into a set builder form first we should know the procedure of it. There are different interval notations for every notation like open, closed or half-open intervals. But the interval given to us is an open interval so for this the general notation is $\\{x : x \in R , a < x < b \\}$ .By using this we can find the set-builder form of the given interval. Here we are taking x $\in$ R because an interval is a set of real numbers.

Complete step-by-step solution:
In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. In other words, the method of defining a set by describing its properties rather than listing its elements is known as set builder form. We can write the given interval of a set in set builder form by using the required mathematical symbols and assuming a variable in each set.
The interval given to us is $\left( {6,12} \right)$ .Consider a symbol x to describe the element of the set or interval which is followed by a colon (:) .After the sign of colon, we write the characteristic property possessed by the elements of the interval and then enclose the whole description within the braces.
An interval is a set of real numbers, all of which lie between two real numbers. Should the endpoints be included or excluded depends on whether the interval is open, closed or half-open. The given interval is of the form (a,b). Therefore its general notation will be $\\{x : x \in R , a < x < b\\}$.
So, now let’s change the given interval into a set builder form. The given interval is $\left( {6,12} \right)$ ,as this interval is open and we know that interval is a set of real numbers. **Therefore, the interval $\left( {6,12} \right)$ in set builder form can be written as
\left( {6,12} \right){\text{ }} = {\text{ }}\left\\{ {x:x \in R,{\text{ }}6 < x < 12} \right\\} **

Note: Remember that the colon (:) stands for ‘such that’ . Keep in mind that an open interval does not include its endpoints and is enclosed in parentheses. Note that a < x < b means all real numbers between a and b, not including a and b.