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Describe the following sets in set-builder form: \[\text{E=}\left\{ 0 \right\}\]

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Last updated date: 15th Jun 2024
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Answer
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Hint: Before solving the problem, we should know about the set-builder form. In a set builder form, the statement is written within the pair of brackets so that the set is well defined. In the set builder form, all the elements of the set must possess a single property to become the member of that set. In this form of representation of a set, the element of the set is described by using a symbol ‘x’ or any other variable followed by a colon The symbol ‘:‘ or ‘|‘ is used to denote such that and then we write the property possessed by the elements of the set and enclose the whole description in braces. In this, the colon stands for ‘such that’ and braces stand for ‘set of all’. By using this concept, we should write the set-builder form of \[\text{E=}\left\{ 0 \right\}\].

Complete step-by-step answer:
Before solving the problem, we should set the builder form. In this, the statement is written within the pair of brackets so that the set is well defined. In the set builder form, all the elements of the set must possess a single property to become the member of that set. In this form of representation of a set, the element of the set is described by using a symbol ‘x’ or any other variable followed by a colon The symbol ‘:‘ or ‘|‘ is used to denote such that and then we write the property possessed by the elements of the set and enclose the whole description in braces. In this, the colon stands for ‘such that’ and braces stand for ‘set of all’.
So, now by using this concept we should write the \[\text{E=}\left\{ 0 \right\}\]in set-builder form. From this set, it is clear the element in the set is 0. Now let us write this in set-builder form.
\[E=\left\{ x:x=n-n;n\in R \right\}.....(1)\]
The equation (1) represents the set-builder form of \[\text{E=}\left\{ 0 \right\}\] because for every value of x, we get the value of every element is equal to zero.

Note: Students may think that the set builder form of \[\text{E=}\left\{ 0 \right\}\]is \[E=\left\{ x:x=0 \right\}\]. But this is a wrong form of writing a set-builder form. Students should write the set-builder form as mentioned above and this misconception should be avoided to have a correct solution and a correct answer. Students should have a clear view of the concept.