Set and Representation of a Set

Set and Representation of a Set

## Definition / Concept

A set is a well-defined collection of distinct objects. The objects belonging to a set are called elements or members.

## Formula

$$A=\{a_1,a_2,a_3,\ldots,a_n\}$$

$$A=\{x:x\text{ satisfies a given property }P\}$$

## Terminologies

- Element → An object belonging to a set.

- Roster form → Representation by explicitly listing elements.

- Set-builder form → Representation using a defining property.

- Universal set → Set containing all objects under consideration.

## Conditions / Special Cases

Order of elements does not matter:

$$\{1,2,3\}=\{3,2,1\}$$

Repetition of elements does not create new elements:

$$\{1,1,2,2,3\}=\{1,2,3\}$$

## Usage

- Used to represent collections of objects and form the basis of set theory.