Question

If A=\left\\{ x:x\in N\text{ and x20} \right\\} and B=\left\\{ x:x\in N\text{ and x}\le \text{5} \right\\} then write the set $A-B$ in the Set-Builder form.

Answer 1

In the given question, we have the sets A and B given in set builder form. We need to convert this set builder form for both into a form with a set containing the elements. This is known as the roster form. We then perform the operation of set subtraction and write the final answer again in terms of set builder form.

Complete step-by-step answer:
For the given question, let us convert the given set builder forms for both A and B into a set of elements. Consider the set A, A=\left\\{ x:x\in N\text{ and x20} \right\\}. This means that we need to select the set A such that all the elements inside it are natural numbers less than 20. N here in the equation represents natural numbers and we know that natural numbers start from 1 and extend up to infinity. Representing the above equation in set of elements form, the roster form,
\Rightarrow A=\left\\{ 1,2,3,4\ldots ,18,19 \right\\}
Similarly, we need to represent set B in a set of elements. Given B, B=\left\\{ x:x\in N\text{ and x}\le \text{5} \right\\}, set B is selected such that all the elements inside it are natural numbers less than or equal to 5.
\Rightarrow B=\left\\{ 1,2,3,4,5 \right\\}
Set subtraction means that we need to remove all the elements from A that are contained in B. Hence, removing all the elements of B from A,
\Rightarrow A-B=\left\\{ 6,7,8,9\ldots 18,19 \right\\}
This above equation is in the roster form or the form of a set of elements, we need to convert this to the set builder format. This can be written in set builder format as,
\Rightarrow A-B=\left\\{ x:x\in N\text{ and 5x20} \right\\}
This can also be written using less than or equal to and greater than or equal to as,
\Rightarrow A-B=\left\\{ x:x\in N\text{ and 6}\le \text{x}\le 19 \right\\}
The above two equations represent the solution. Hence, we have written the set $A-B$ in set builder form for the given sets A and B.

Note: It is important to know the concept of sets to solve this question. We need to know that a set in general can be represented in two ways. One is the roster form or tabular form which lists each element of the sent in a pair of curly braces and the other is a set-builder format in which the notation is given in words.