Set Builder Notation Math

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Introduction to set builder notation in math:

In set theory and its applications to logic, mathematics, and computer science, set-builder notation (sometimes simply set notation) is a mathematical notation for describing a set by stating the properties that its members must satisfy. In math, forming sets in this manner is also known as set comprehension, set abstraction or as defining a set's intension. (Source: From Wikipedia).

Explanation of set builder notation in math:

The common property of set should be such that it should specify the objects of the set only. For example, let us consider the set {6, 36, 216}.
The elements of the set are 6, 36 and 216. These numbers have a common property that they are powers of 6. So the condition x = 6n, where n = 1, 2 and 3 yields the numbers 6, 36 and 216. No other number can be obtained from the condition.
Thus we observe that the set {6, 36, 216} is the collection of all numbers x such that x = 6n, where n = 1, 2, 3. This fact is written in the following form {x | x = 6n, n = 1, 2, 3}. In words, we read it as the set consisting of all ...
... x such that x = 6n, where n = 1, 2, 3.
Here also, the braces { } are used to mean ‘the set consisting of ’. The vertical bar ‘ | ’ within the braces is used to mean ‘such that ’. The common property ‘x = 6n, where n = 1, 2 and 3 acts as a builder for the set and hence this representation is called the set–builder or rule form.
If P is the common property overcome by each object of a given set B and no object other than these objects possesses the property P, then the set B is represented by { x | x has the property P} and we say that B is the set of all elements x such that x has property P.

Problems in set builder notation:

Example problem 1:

Represent the following sets in set builder notation:

(i) The set of all natural numbers less than 8.

(ii) The set of the numbers 2, 4, 6, … .

Solution:

(i) A natural number is less than 8 can be described by the statement:

x ? N, x < 8.

Therefore, the set is {x | x ? N, x < 8}.

(ii) A number x in the form of 2, 4, 6, … can be described by the statement:

x = 2n, n ? N.

Therefore, the set is {x | x = 2n, n ? N}.

Example problem 2:

Find the set of all even numbers less than 28, express this in set builder notation.

Solution:

The set of all even numbers less than 28.

The numbers are, x = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26}

{x | x is a even number, x < 28}.

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