Question

# Describe the following set in Roster form:$\left\{ x\in N:x=2n\text{,}n\in N \right\}$

Hint: In this question, we are given a set in the set builder form and we are asked to describe the same in the roster form. Therefore, we should first understand the roster method of representing a set and then find out all the elements of the set which satisfy the given condition. Thereafter we can use these elements to write the given set in the set builder form.

In this method we have to convert the set from set builder form to Roster form. Therefore, we should first understand the definition of the set builder and roster form which are as follows:
a)In the set builder form, the property which an element in the set satisfies is specified by writing a variable followed by a colon and then the property satisfied by the variable. Thus, the variable can take all the values which satisfy the property and thus all such values of the variables will be elements of the set. For example, in the set
{x: property of x}
all values of x which satisfy the given condition will be part of the set……………… (1.1)

b)In the roster form, all the elements of the set are written explicitly within curly braces with the elements separated by a comma. For example, if a, b, c, d and e are elements of the set A, then it can be represented as
A={a,b,c,d,e}………………………… (1.2)
In this question, if we name the given set to be S, the it is represented as
$S=\left\{ x\in N:x=2n\text{,}n\in N \right\}..................(1.3)$
Now, we know that the set of natural numbers is given by
$N=\left\{ 1,2,3... \right\}...........(1.4)$
Now, we see that all the numbers of N multiplied by 2 will be elements of the set S. Thus, taking $n=1,2,3...$, the elements of $S$ would be $2\times 1,2\times 2,2\times 3,....$. i.e. $\left\{ 2,4,6,... \right\}$ upto infinity as the set N itself contains infinite number of elements.
Therefore, using the definition (1.2), we can write the given set in roster form as
$\left\{ x\in N:x=2n\text{,}n\in N \right\}=\left\{ 2,4,6... \right\}$
Thus, {2,4,6,...} is the description of the set in roster form and is the answer to this question.

Note: We should note that in this case even though in the set builder form, the property could be written very precisely, the actual number of elements contained in the set which appear in the roster form is infinite. However, it is not having any problem as a set can contain an infinite number of elements.