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What is Set Builder Notation? (with Examples)

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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.

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,….}.

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 ∈ N | x ≥ 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. N – The set of natural numbers 1, 2, 3, 4,….so on.
  2. W – The set of whole numbers 0, 1, 2, 3, 4,….so on. Basically, whole numbers consist of zero and natural numbers.
  3. Z – The set of integers = {….,-3, -2, -1, 0, 1, 2, 3, 4,….}
  4. R – The set of all real numbers.
  5. ∈ – 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 ∈ N | x ≥ 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 ∈ N | 12 ≤ x ≤ 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={Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.
    • This set can be represented in set builder notation as, A={ x | x 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 interval [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 ∈ R∣ axb}.
  2. The interval (a,b] consists of all real numbers between a and b excluding a and including b.
    • It can be represented in set builder form as, (a,b] = {x ∈ R ∣ a < xb}.
  3. The interval [a,b) consists of all real numbers between a and b including a and excluding b.
    • It can be represented in set builder form as, [a,b)={x ∈ R ∣ ax < b}.
  4. The interval (a,b) consists of all real numbers between a and b excluding a and b themselves.
    • It can be represented in set builder form as, (a,b)={x ∈ R ∣ a < x <b}.

Summary
Article Name
What is Set Builder Notation? (with Examples)
Description
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.

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