Basics and applications of Set Builder Notation

When detailing the components of a set or demonstrating the criteria, the sets’ members must match. Mathematicians use it as a descriptive term for groups of elements or to illustrate the features of a group.

So, let’s learn more about set builders with set builder notation examples.

Method of writing sets as per set-builder notation

You can use the ellipses notation to describe which items are included in a set, even though there is no order among the members. You can do this by including an ordered sequence before (or after) the ellipses, which serves as a convenient notational vehicle. Initially, the sequence’s first few components are shown. Then the ellipses indicate that the simplest interpretation should be adopted to go on with the series in the following components of the sequence. If there is no final value appearing to the right of the ellipses, it is considered that the series is unbounded.

Definition of set builder notations:

• Using set-builder notation, you may describe the properties of a set without identifying all of its parts. This is useful for defining collections of things.
• Set-builder notation’s process of constructing a set is described by terminology like set abstraction, set comprehension, and set purpose.
• In the set-builder notation, a variable or variables are provided and a rule that determines which objects get into the set and which ones don’t. 
• In formal language, predicates are often used to demonstrate this rule. The vertical slash “|” or the colon denotes the set rule and the variables (:). Using this method is common for describing infinite sets.

For example, {y: y > 10} is read as: “the set of all y’s, such that y is greater than 10”.

Set in mathematics

An unordered collection of items represented by the sequence of elements (separated by commas) between curly braces “{ and “} is known as a set in mathematical terms.

{ cow, Cat, and dog} are examples of domesticated animals; {9, 3, 5, 7, and 11} are examples of odd numbers; and letters {d, e, f, g, and h} are examples of alphabets.

Set-builder notation examples

X = {y: y is a letter between a to f}

We read it as,

“X is the set of all y such that y is a letter between a to f”.

Set Builder Notation Symbols

The different symbols that are used to represent set builder notation are as follows:

• The letter Z represents integers
• All positive integers are represented by the symbol N
• The letter R denotes numbers that are not fictitious
• The letter Q stands for either rational numbers or fractional numbers
• The whole number is represented by the symbol W
• The sign ∈ denotes “is a part of” the sentence
• The sign ∉ “is not a part of” the sentence.

Set Builder Notation Examples in Details

Sets Methods

There are two different methods to represent sets. These are:

1. Tabular Form or Roasted Method.
2. Set -Builder Form or Rule Method.

Tabular form:

In the roasting approach, the set components are listed between the braces, with each element being separated by a comma between each element in the list. If an element occurs more than once in the collection, it can only be written once per instance of the element.

Example,

The set A of the letter of the word CHENNAI is written as A = {C, H, E, N, A, I}.

The set X of the first eight natural numbers is written as X = {1,2,3,4,5,6,7,8}.

as we can see, The elements of the set in the roasted method can be listed in any order. Hence, the set {A,B,C,D,E} can be written as {C, A, B,D, E}.

Set -Builder Form

If the components in a set have a common property, the set elements may be described by describing the standard feature. 

Consider the set A = {1,2,3,4,5,6,7} which has a common feature that all of the members in the set A are natural numbers less than 8. This property indicates that all of the elements in set A are natural numbers less than 8. Any other natural numbers do not share this trait. As a result, the set X may be written as follows:

We can also write, set A = {the set of all the natural numbers less than 8}.

A = {x: x is a natural number less than 8} which can be read as “ A is the set of elements x such that x is natural numbers less than 8”.

The above set can also be written as A = {x : x N, x < 8}.

Meaning of unordered mean in the set

In mathematics, sets are not arranged logically or sequentially. The set X = {1, 2, 3, 4,5} 5 seems to be the set of ordered numbers between 1 and 5, but it is identical to the set Y = 5, 4, 2, 3, 1 (which appears to be the set of ordered numbers between 1 and 5). It makes no difference what order items are arranged in a set. Two sets are considered to be equal if they include all of the items in each of them.

Conclusion

It is possible to define a set by enumerating its components or by stating the attributes that each of its members must possess in a mathematical language known as set-builder notation. It is used in various domains, including set theory and its applications to logic, mathematics, and computer science, among others.