Question

Write the following sets in set-builder (Rule method) form: \[{B_2} = \left\{ {11,13,17,19} \right\}\].

Answer 1

Hint: Here, we will identify a common property/rule shared by the elements of the given set, and then use that rule to write the set in set-builder form. Set-builder form is used to represent a set the elements of which share a common property.

Complete step by step answer:
A set is a collection of objects that are well defined. Sets can be represented by two methods: Roster form or set-builder form.
In roster form, all the elements of a set are written within the brackets \[\left\{ {} \right\}\], separated by commas.
Set-builder form is used to represent a set the elements of which share a common property or rule. Writing a set in this form is known as the rule method of describing sets.
To convert a set from roster form to set-builder form, we need to find a rule or property that is shared by all the elements of the set.
Now, the given set is \[{B_2} = \left\{ {11,13,17,19} \right\}\].
The elements of the set are 11, 13, 17, and 19.
We need to find a rule that is shared by all the elements of the set.
First, we assume a number \[x\].
Now, we decide the range of values which \[x\] can take.
Let \[x\] lie between the integers 10 and 20, both exclusive.
\[10 < x < 20\]
Here, \[x\] can take the values 11, 12, 13, 14, 15, 16, 17, 18, and 19.
Next, we identify a rule that is followed by only the numbers 11, 13, 17, and 19.
Assume that \[x\] is not divisible by either 2, or 5.
The numbers 12, 14, 15, 16, and 18 are either divisible by 2 or 5. These cannot be the value of \[x\].
Thus, the rule shared by 11, 13, 17, and 19 is that they lie between 10 and 20, and are not divisible by either 2 or 5.
Now, we can write the given set in set-builder form.
\[{B_2} = \left\{ {x:x{\text{ is not divisible by 2 or 5}};10 < x < 20} \right\}\]


Note:
When converting a set from roster form to set-builder form, multiple answers are possible. Suppose that \[x\] is a number representing a prime number lying between 10 and 20, both exclusive. The possible values of \[x\] are 11, 13, 17, and 19, that is all the elements of the set \[{B_2}\]. Thus, the set \[{B_2}\] can also be written as \[{B_2} = \left\{ {x:x{\text{ is a prime number}};10 < x < 20} \right\}\] in set-builder form.