Among the different types of sets, finite and infinite sets are the two that are generally considered while dealing with mathematics. As per those it could be considered as a way to group similar objects.
Sets
Any considerable collection of mathematical objects that is well-defined can form a set. These objects could probably be anything like people’s names, ages, the things they like or dislike. It could comprise of complex scientific data or entities from the simplest number system and even the outcomes from 100s or 1000s of times repeated experiments from a coin toss or a dice roll. The only factor that holds importance is that the objects or entities should be related through a similar rule. Mentioned below are several examples for the same:
- A collection of the names of Indian freedom fighters.
- The family of natural numbers/odd numbers/while numbers/even numbers/rational numbers/ real numbers/integers.
- A group of possible outcomes from a coin toss or a dice roll.
- A collection of crucial that may be collected from MOM or ISRO.
- The night/day temperatures for a considerable period.
Representations of Sets
Representation of sets could be done in two ways:
- Roster form
- Set builder form
Roster Form
In this form, all the elements of the set are said to be enclosed within {} (braces) and are separated using comma (,). For instance, consider a collection that has the numbers found on a dice M= {1, 2, 3, 4, 5, 6}.
Properties of a roster form
- The order in which the elements are listed in a roster form is for any considerable set is said to be immaterial. For instance, V = {u, o, e, a, i} is similar to V = {a, e, i, o, u}.
- Dots at the end of the elements of any respective set would determine that it’s infinite and indefinite in nature. For instance a set including all odd numbers V={1, 3, 5, …}.
- In this form of representation, elements generally are not repeated.
Set Builder Form
In this form for representation, the elements within the set possess one common factor that isn’t possessed by elements outside the set. For instance, the series of vowels in the English alphabets. If V represents the collection of vowels, then V = {x: x is a vowel in alphabetical series of English language.}
Properties of Set Builder Form
- For this type of representation, a colon (:) is considered to be a mandatory symbol.
- After placing the colon, the common characteristic that is possessed by all the elements is to be mentioned and then enclosed within braces.
- If the set does not follow a pattern, it could not be represented into a set builder form
The above-discussion clarifies the concept of sets and probable representation for those.