Question

Describe the following set in Roster form:
$\left\{ x\in N:x\text{ is a prime number,10x}<20 \right\}$

Answer 1

Hint: In this question, we have been given a set in the set builder form and we have been 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.

Complete step-by-step answer:
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
$S=\left\{ x\in N:x\text{ is a prime number,10x}<20 \right\}.............(1.3)$
Now, we know that a number is called a prime number if it does not have any other factor than 1 and itself, therefore the prime numbers lying between 10 and 20 are 11,13,17 and 19………..(1.4)

Therefore, using equations (1.3) and (1.4), we find that x can take only the value 11,13,17 and 19.
Therefore, using the definition (1.2), we can write the given set in roster form as
$S=\left\{ x\in N:x\text{ is a prime number,10x}<20 \right\}=\left\{ 11,13,17,19 \right\}$
Thus, {11,13,17,19} is the description of the set in roster form and is the answer to this question.

Note: In equation (1.3), we should note the inequality is strict i.e. the number should be strictly less than 20 and not less than or equal to 20. Thus, in the roster form, we should be careful not to include the number 20 as an element. However, in this question as 20 is not a prime number 20 will not get included in the set. However, in other questions, we should always be careful if the inequality is strict or not.