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Planets of the Solar system The collection of the planets of the Solar system is a set because there are exactly eight planets in our Solar system - Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune. Hence, this collection has well defined and distinct elements.
It may be noted that, in the late 1990s, Pluto was excluded from the list of planets as it was considered only as a dwarf planet.
Sets

Definition: A collection of well defined and distinct objects is called a set.
Some examples of sets are: the collection of letters of the English alphabet; the collection of official cities in United States; the collection of all countries in the European Union.
Sets are usually denoted by uppercase letters i.e, in capitals.

Each object belonging to the set is called its element or member.
For example, in the set of letters of the English alphabet, each of the letters a, b, c, d, e, ... , x, y, z will be known as its element or member.
Elements are usually denoted by lowercase letters.

The symbol ∈ (epsilon), a Greek Alphabet is used to indicate that an object is an element of a set or belongs to a set.
For example, if the number ‘1’ is an element of the set P, it will be written in symbols as: 1 P.
The symbol ∉ (crossed epsilon) is used to indicate that an object is not an element of a set or does not belong to a set.
Ex: If the number ‘2’ is not an element of the set P, it will be written in symbols as: 2 P.

Sets representations

There are two ways to represent a set:
(i) Roster form
(ii) Set builder form

Roster form

It is also known as tabular form or list form. In this form, all the elements belonging to the set are listed and are enclosed within curly braces {} after separating them by commas.
Ex: If ‘P’ denotes the set of all the vowels of English alphabet, then ‘P’ can be written in roster form as:
P = {a, e, i, o, u}

In this form, the order of listing the elements is not important. Therefore, we can write the above set as:
{a, e, i, o, u}, {a, i, e, o, u}, {o, u, i, a, e}, etc.
There are many sets which contain a large number or infinite many elements. These types of sets cannot be represented by roster form.

Note: It may be noted that while writing the set in roster form an element is not generally repeated, i.e., all the elements are taken as distinct.
Ex: The set of letters forming the word 'BANANA' is {B, A, N}

Set builder form

It is also known as rule form. In this form, we define the property or the rule satisfied by all the elements of a set.
Ex: If ‘P’ denotes the set of all the vowels of English alphabet, then ‘P’ can be written in the set builder form as:
P = {x : x is a vowel of English alphabet} or P = {x | x is a vowel of English alphabet}

There are many sets which contain a large number or infinite many elements. These types of sets can be represented by set builder form.

Types of Sets

Sets are classified into different types according to the elements they have. Some of these types are given below:

  • Finite set: A set that contains finite number of elements is called a finite set. The elements of a finite set can be counted.
    Ex: The set of natural numbers less than 100; the set of months in a year; the set of solutions of a quadratic equation; etc.

    The number of elements in a finite set is called its order or cardinal number or cardinality of the set.
    Ex: The cardinality of set of vowels of English alphabet {a, e, i, o, u} is 5 because it contains 5 elements.
    If ‘A’ is any finite set, then its cardinal number is denoted by n(A).

    The cardinal numbers for the three examples given above for finite sets are:
    n(the set of natural numbers less than 100) = 99;
    n(the set of months in a year) = 12;
    n(the set of solutions of a quadratic equation) = 2
  • Infinite set: A set that is not a finite set is termed an infinite set. Infinite sets may be countable or uncountable.
    Ex: (i) the set of all integers, {. . . . , -1, 0, 1, 2, . . . . }, is a countably infinite set;
    (ii) the set of all real numbers and the set of all points (or lines) in a plane are uncountably infinite sets.
  • Empty set: A set which does not contain any element is called an empty set or null set or void set. It is denoted by Greek alphabet ‘Φ’ (Phi) or {} without any element in the braces.
    Ex: the set of natural numbers between – 2 and – 5; the set of integers between (1/2) and (1/4); the set of days between Friday and Saturday; etc.
  • Singleton set: A set that contains only one element is called a singleton set.
    Ex: the set of whole numbers between 0 and 2 which is {1};
    the set of multiples of 5 between 8 and 11 which is {10};
    the set of factors of 6 between 4 and 23 which is {6}; etc.
  • Non-empty set: A set that contains at least one element is called a non-empty set.
    Ex: the set of natural numbers between 1 and 40; the number of people on earth; the set of multiples of 6 between 8 and 80; etc.
  • Universal set: A set that contains all the elements (objects) of the sets under consideration is called a universal set. It is usually denoted by U.
    Ex: For "the set of all even numbers", the universal set can be "the set of real numbers".