Question

Write the following set in set – builder (Rule method) form: - ${{B}_{5}}$ = {-5, -4, -3, -2, -1}.

Answer 1

Read the definition of the set and its two types, that is set – builder form and roster form. Check out the properties of these forms and the difference between them to write the set given numbers in set – builder form. Assume the given numbers as a single variable x and define the range of x from -5 to -1.

Complete step-by-step solution
Here, we have to represent the set of numbers in ${{B}_{5}}$ in set – builder form. Let us first see the definition of set and its type of representation.
In mathematics, a set is a well–defined collection of distinct objects. The objects that make up a set are known as the set’s elements or members. These elements can be anything: numbers, people, letters of the alphabet, other sets, and so on. There are two ways of representing members of a set.
1. Roster form: -
The roster notation method of defining a set consists of listing each member in the set. In this form, the set is denoted by enclosing the list of members in curly brackets: -
A = {1, 9, 10, 15, 60}
Here, we can write the numbers in any order.
2. Set – builder form: -
In the set-builder form, the set is specified as a selection from a large set, determined by a condition involving the elements. For example, a set A can be specified as follows: -
A = {x : x is an integer, and $16\le x\le 20$}
Here, the colon (:) means ‘such that”.
Now, let us come to the question. We have been provided with a set ${{B}_{5}}$ which is represented in roster form. We have to convert it into set – builder form.
Since, ${{B}_{5}}$ = {-5, -4, -3, -2, -1} represents integers from -5 to -1. So, let us assume these integers as a variable ‘x’ whose range is from -5 to -1. Therefore, in set – builder form we have,
$\Rightarrow {{B}_{5}}$ = {x : x is an integer, and $-5\le x\le -1$}

Note: One must note that we have to use curly braces { } for the representation of sets. If we will use small brackets ( ) or square brackets [ ] in set – builder or roster form then it will be a wrong notation. Do not forget to write the condition that the variable ‘x‘ represents and its range, otherwise the set will be meaningless.