Set-Builder Notation

Set-builder Notation is a type of mathematical notation used to describe sets by naming their components or highlighting the requirements that each member of the set must meet. Sets are written in the form of {y | (properties of y)}  OR {y : (properties of y)} in the set-builder notation, where the condition that fully characterizes each member of the collection replaces the attributes of y.

The elements and properties are separated using the character ‘|’ or ‘:’ The entire set is interpreted as "the set of all elements y" such that (properties of y), while the symbols ‘|’ or ‘:’ are read as "such that."

This article explores the set-builder notation, symbols used in set-builder notation, examples, representation of sets methods, etc.

What is Set-Builder Notation?

A representation or notation known as "set-builder notation" is used to express a set that is defined by a logical formula that simplifies to be true for each element of the set. There may be one or more variables included. It also specifies a rule for the set's constituent members.

Symbols Used in Set Builder Notation

The elements of the set are represented by a variety of symbols in the set builder form. Here is a list of some of the symbols.

• | stands for "such that" and is often inserted after the variable in the set builder form. The set condition is then written after this symbol.
• ∈ When translated as "belongs to," or in other words "is an element of".
• The word, ∉ when translated as "does not belong to," implies "is not a part of."
• The letter N stands for all positive integers or natural numbers.
• W stands for whole numbers.
• Z stands for integers.
• Any number that may be stated as a fraction of integers or as a rational number is represented by Q.
• Any number which is not rational is called Irrational Number and is represented by P.
• R stands for real numbers or any non-imaginary number.
• C stands for Complex Numbers.

Representation of Sets Methods

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

• Tabular Form or Roster Form
• Set-Builder Form or Rule Method

Read More: Representation of Sets

Tabular or Roster Form

The items of the set are enumerated using the roasting method's braces{}, with commas between each piece. The element can only be written once if it appears more than once in the collection.

Examples of Roster Method

• The formula for the first five natural integers, designated as set X, is X = {6, 7, 8, 9}
• The letter combinations {D, E, L, H, I} make up the set A of the word DELHI.

The Roster Method is another name for this approach of defining sets. Any order may be used to list the components of the roster set. As a result, the set {A, B, C, D} may be expressed as {B, A, C, D}.

Set-Builder Notation

If a set's components share a property, that property can be used to define the components. For instance, the set A = {1, 2, 3, 4, 5, 6} has a trait in common that all of its members are natural integers lower than 7. Other natural numbers do not have this characteristic. As a result, the set X may be expressed as follows:

A = {x: x is a natural number less than 7} may be translated as "A is the set of elements x such that x is a natural number less than 7."

The set mentioned above may alternatively be expressed as A = {x: x N, x < 7}.

Another way to express set A = {the set of all natural numbers less than 7}.

In this instance, the description of a set's common attribute is written inside brackets. This is a set-builder form or rule approach in its most basic version.

Why Do We Use Set Builder Form?

When there are many components and utilizing the roster form makes it difficult to represent the components of the set, set builder notation is employed. Let's use an illustration to better grasp this. If you need to write a list of numbers from 1 to 8 inclusively, you may just write {1, 2, 3, 4, 5, 6, 7, 8} using the roster notation.

But when we have to list every real number, a difficulty appears. It would not be possible to use roster notation in this situation. {..., 1, 1.1, 1.01, 1.001, 1.0001, ... }. However, in this case, it would be preferable to use the set-builder notation.

{x | x is a real number} OR {x | x is a rational or irrational number} is the set builder form for real numbers. Writing sets are made easier by using the set-builder notation, particularly when writing sets with an unlimited number of components. The set-builder notation may be used to express numbers like integers, real numbers, and natural numbers. Using this technique, a set containing an interval or an equation may also be represented.

How to use a Set Builder Notation?

A mathematical notation known as "set builder notation" lists all the requirements that each member of a set must meet in order to be included in the set. In particular, it helps to understand sets with an unlimited number of items.

There are three key parts to set builder notation:

• Typically, a variable is written in lowercase.
• "Such that" is viewed as a vertical bar separator or colon.
• logical statement of the characteristics of sets.

The three elements of set builder notation mentioned above are placed inside curly brackets as shown below:

A = { variable | attribute }

OR

A = { x : θ(x) }

A separator, the vertical bar is either interpreted as "such that" or a colon “:”. θ(x). For all values of x for which the predicate is true, the set being defined is represented by the symbol (x), which corresponds to the predicate (a logical statement indicating the attributes that the set contains).

How to Write a Set Builder Notation?

Let's go on to the next idea, writing the set-builder notation, now that we understand what that is.

To write sets in set builder notation, follow the instructions below:

• Use a lowercase letter, such as x, or any other letter, to denote the components of a set.
• As a divider, use a colon (:) or vertical bar (|).
• Declare the property's requirement that every member of the supplied set has items after the symbol.
• Inside the curly braces{}, type the whole description.

Think about the following illustration where set A is described as:

A = { x ∈ R | x<4 }

Where the symbol ∈ means "member of." The abbreviation "R" stands for "real numbers." Because x in R can be any number less than 4, set A holds the value of x in R.

How to read Set Builder Notation?

A technique for expressing set attributes that hold true for each and every element contained in the set is called set builder notation. The format of set builder notation is as follows:

A = { x | condition about x }

is to be understood as "the set of all the values of x such that the given condition about x is true for all the values of x."

A vertical bar can be used in lieu of the colon and is interpreted in the same way.

A = { x : condition about x }

The words "such that" in the set-builder notation explanation are represented by the colon and the vertical bar, respectively.

Set Builder Notation for Domain and Range

Writing the domain and range of a function using the set builder notation is quite helpful. The set of all the values that are input into a function is the domain of the function. For example, the domain of the rational function f(x) = 2/(x-1) would include all real integers other than 1. This is due to the fact that when x = 1, the function f(x) would be undefined. As a result, the domain of this function is written as  {x ∈ R | x ≠ 1}.

The set builder notation may also be used to indicate the range of a function. The range of the function is a set of the values that a function can take and for the function f(x) = 2/(x-1) we define the range as,

y = 2/(x-1)

⇒ x - 1 = 2/y

⇒ x = 2/y + 1,

Thus we define the range of function, in the set builder notation as, {y ∈ R | y ≠ 0}.

People Also Read:

• Representation of a Set
• Types Of Sets
• Operations on Sets
• Venn Diagrams
• Subset
• Universal Set

Set Builder Notation Examples

Example 1: Use the set-builder notation to represent the given set.

A = {2, 4, 6, 8,10}

Solution:

Provided set A = {2, 4, 6, 8,10}  in the set-builder form is represented as:

{x: x is an even natural number less than 12"}.

Example 2: How should x ≤ 3 or x ≥ 4 be written in set-builder notation?

Solution:

In set builder notation, x ≤ 3 or x ≥ 4 may be written as:

{x ∈ R | x ≤ 3 or x ≥ 4}

Example 3: Use interval notation to represent the set that contains all positive real values.

Solution:

The number that is bigger than 0 would serve as the starting point for the set of positive real numbers, albeit we are unsure of the precise value of this number. Positive real numbers also exist in an unlimited number of combinations. As a result, we may express it as the interval (0, ∞).

Example 4: Decode the symbols that are presented:

(i) 5 ∈ ℚ

(ii) -8 ∉ ℕ 

Solution:

Set of natural numbers is N, while the set of rational numbers is Q.

(i) 5 ∈ Q denotes that 5 is a member of a group of rational numbers.

(ii) -8 ∉ N indicates that -8 does not fall under the category of natural numbers.

Summary - Set-Builder Notation

Set-builder notation is a mathematical shorthand used to define sets based on specific properties that all elements of the set share. It is particularly useful when dealing with large or complex sets where listing all elements individually would be impractical or impossible. In set-builder notation, you typically start with a variable that represents the elements of the set, followed by a vertical bar (|) or colon (:), which can be read as "such that." After this separator, you describe the condition or rule that all elements must satisfy to be included in the set.

This method is extremely versatile and can be applied to any collection of numbers or objects that fit a certain rule, from simple sets like natural numbers under a certain value, to more complex sets involving equations or inequalities. The notation is compact and precise, making it a powerful tool for conveying a lot of information about a set in a very succinct way. Whether it's specifying the domain and range of a function, or defining intervals of real numbers, set-builder notation simplifies the expression of mathematical ideas and ensures clarity and precision in mathematical communication.