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# Set Builder Notation – Explained with Examples

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The set builder notation is one of the methods that is used to represent sets. The other method used to describe the elements of a set is the roster form. Suppose we are given a set, A=\{ 3, 4, 5, 6, \ldots \}.

The set shown above is written in roster form. This means that we simply list the elements of the set. The dots after 6 mean that the set contains all natural numbers greater than 6. We can represent this set in set builder notation as follows, A=\{ x\in \mathbb{N} | x \geq 3 \}. In order to understand what the above notation means we must understand the meaning of the symbols used in the above notation.

### Symbols used in Set Builder Notation:

Some of the most common symbols used when writing sets in set builder form are:

1. \mathbb{N} – The set of natural numbers 1, 2, 3, 4,….so on.
2. \mathbb{W} – The set of whole numbers 0, 1, 2, 3, 4,….so on. Basically, whole numbers consist of zero and natural numbers.
3. \mathbb{Z} – The set of integers = {….,-3, -2, -1, 0, 1, 2, 3, 4,….}
4. \mathbb{R} – The set of all real numbers.
5. \in – This symbol is read as “belongs to”.
6. | – This symbol is read as “such that”.

### Examples of Set Builder Notation:

1. The example A=\{ x\in \mathbb{N} | x \geq 3 \} given above is read as – “The set of all x belonging to natural numbers such that x is greater than or equal to 3”. This is precisely the set of numbers {3, 4, 5, 6,…so on}.
2. Consider the set A=\{ 12, 13,...., 27 \}. This set can be represented in set builder notation as, A=\{ x\in \mathbb{N} | 12\leq x \leq 27 \}. The above notation is read as – “The set of all x belonging to natural numbers such that 12 is less than or equal to x and x is less than or equal to 27”.
3. Consider the set A=\{ \text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} \}. This set can be represented in set builder notation as, A=\{ x | x \text{ is a day of the week} \}. The above notation is read as – “The set of all x such that x is a day of the week”.

The general idea when trying to write any set in set builder notation is that we should identify a rule which describes the elements of the set. This “rule” appears in the notation after the “such that” (|) symbol.

#### Interval notation to set builder notation:

We have the following four different kinds of intervals which can be written in set builder form as follows:

1. The inteval [a,b] consists of all real numbers between a and b including a and b themselves. It can be represented in set builder form as, [a,b] = \{x\in \mathbb{R}| a\leq x \leq b\}.
2. The inteval (a,b] consists of all real numbers between a and b excluding a and including b. (a,b] = \{x\in \mathbb{R}| a < x \leq b\}.
3. The inteval [a,b) consists of all real numbers between a and b including a and excluding b. [a,b) = \{x\in \mathbb{R}| a\leq x < b\}.
4. The inteval (a,b) consists of all real numbers between a and b excluding a and b themselves. (a,b) = \{x\in \mathbb{R}| a < x < b\}.

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