A notation that expresses a set constructively by defining the properties of members (or elements) in a mathematical rule is called the set-builder notation.

Mathematics is working on the basis of rules or formulas. So, the idea had come to express a set constructively in a special mathematical notation by defining the properties of elements (or members) in a formula. Hence, the constructive representation of a set in a mathematical rule is called the set-builder notation.

The set-builder notation consists of the following three parts.

- A variable
- A Separator
- A formula

Now, let’s learn each part in the set-builder form understandably.

A variable is used to represent each member (or element) in a set.

A colon $(:)$ or vertical bar $(|)$ is used to separate the formula from the variable in the notation. The separator is read as “such that” or “for which” or “with the property that”.

A set is usually defined by considering one or more common properties of the elements. It can be expressed by a logical predicate (or formula).

A set-builder notation of any set can be expressed in anyone of the following two forms.

$(1).\,\,\,$ $\Big\{x \,\,:\,\, s(x)\Big\}$

$(2).\,\,\,$ $\Big\{x \,\,\,\,|\,\,\,\, s(x)\Big\}$

In this set, the curly brackets represent a set. Inside the curly braces, a variable, a separator and a logical predicate are written one after one for expressing a set in a mathematical form. In the above form, $x$ is a variable, colon or vertical bar is a separator and $s(x)$ is a formula that the one or more common properties of the objects (or elements).

The roster notation is a simple notation for expressing a set in mathematical form but it has some limitations. Hence, the concept of set-builder notation is introduced in set theory.

Let’s learn the concept of set-builder form from some understandable examples.

The numbers, which are greater than $5$ in set-builder notation.

$(1).\,\,\,$ $A$ $\,=\,$ $\Big\{x \,:\, x \,>\, 5\Big\}$

$(2).\,\,\,$ $A$ $\,=\,$ $\Big\{x \,\,\,|\,\,\, x \,>\, 5\Big\}$

The set $A$ is read as the set of all $x$ such that $x$ is greater than zero.

The numbers, which are greater than or equal to $8$ in set-builder notation. In this example, the type of number is defined. Hence, the set $B$ is written in the following form.

$(1).\,\,\,$ $B$ $\,=\,$ $\Big\{x \,:\, x \,∈ \, N \,\,and\,\, x \,\ge\, 8\Big\}$

$(2).\,\,\,$ $B$ $\,=\,$ $\Big\{x \,\,\,|\,\,\, x \,∈ \, N \,\,and\,\, x \,\ge\, 8\Big\}$

The set $B$ is read as the set of all $x$ such that $x$ is a natural number and the value of $x$ is greater than $8$. The set $B$ is also written in the following form simply.

$(3).\,\,\,$ $B$ $\,=\,$ $\Big\{x \,∈ \, N \,:\, x \,\ge\, 8\Big\}$

$(4).\,\,\,$ $B$ $\,=\,$ $\Big\{x \,∈ \, N \,\,\,|\,\,\, x \,\ge\, 8\Big\}$

The real numbers, which are less than or equal to $1$ and greater than $4$ in set-builder form.

$(1).\,\,\,$ $C$ $\,=\,$ $\Big\{x \,∈ \, R \,:\, x \,\le\, 1 \,\,\,or\,\,\, x \,>\, 4\Big\}$

$(2).\,\,\,$ $C$ $\,=\,$ $\Big\{x \,∈ \, R \,\,\,|\,\,\, x \,\le\, 1 \,\,\,or\,\,\, x \,>\, 4\Big\}$

The set $C$ is read as the set of all $x$, which are real numbers such that the value of $x$ is less than or equal to one or greater than four.

Latest Math Topics

Jun 05, 2023

Jun 01, 2023

May 21, 2023

May 16, 2023

May 10, 2023

Latest Math Problems

Jun 08, 2023

May 09, 2023

A best free mathematics education website that helps students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved