Question

# Describe the following set in set – builder form.$A=\left\{ 1,2,3,4,5,6 \right\}.$

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Hint: The set builder notation of any set $A$ is given as $A=\{x:\text{property of x}\}$.

Here , we mention the property of the elements of the set .
We know that set builder notation is a notation in which the set is described by using the properties of the elements .
For example : Consider the set $B=\{2,4,6,8,10\}$. Here $B$ is a set of elements which are positive and even numbers less than $12$. So , the property of the elements of the set is that they are positive even numbers less than $12$. So , the set $B$ is represented in set builder form as $B =\left\{x:x\text{ is positive even number} < 12\right\}$ .
Now , we are given the set $A$.
The elements of the given set $A$ are $A=\left\{ 1,2,3,4,5,6 \right\}$
We can clearly see that the elements are natural numbers which are less than $7$.
Now , we know that set builder notation is a notation in which the set is described by using the properties of the elements .
So , here in set $A$, we can see the property of the elements is that they are natural numbers less than $7$.
So , in set builder form, we can write the set $A$ as $A=\left\{ x:x\text{ is natural number }<7 \right\}$.

Note: The set builder form is read as “$A$ is the set of all $x$ such that $x$ is a natural number less than $7$”. Students generally get confused between set builder notation of sets and roster notation of sets . Both are different and hence , should not be confused . This confusion can lead to wrong answers .