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Sets and Relations

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.

Introduction to Sets

Definition: A collection of well defined and distinct objects is called a set. Some examples of sets are: the collection of all US Presidents, the collection of letters of the English alphabet, the collection of official cities and towns in United States. Sets are usually denoted by uppercase letters.

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.

Representation of a Set

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

  • Roster form: It can also be 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. For example, 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. For example, the set of letters forming the word 'BANANA' is {B, A, N}.
  • Set builder form: It can also be known as rule form. In this form, we define the property or the rule satisfied by all the elements of a set. For example, 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.
Real life example of a finite set An example for a finite set is: the set of berth numbers in a train compartment. The berth numbers are countable. Say, there are 72 berths in a 3-tier coach and 8 berths per coupe. The set (B) consists of: {1L, 2M, 3U, 4L, 5M, 6U, 7SL, 8SU, ..., 71SL, 72SU}, where L, M, U, SL, SU stand for Lower, Middle, Upper, Side Lower and Side Upper berths. n(B) = 72.

Real life example of an infinite set
The best real life example for an infinite set is: the set of digits in pi (π). The set of digits in pi (π) is uncountable.
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. For example, 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: 99, 12 and 2 respectively.
  • Infinite set: A set that contains infinite number of elements is called an infinite set. The elements of an infinite set can not be counted. Ex: the set of natural numbers, the set of integers, the set of whole numbers, the set of real numbers, the points lying on a plane, etc.
  • 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 {}. 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 natural numbers between 2 and 4, the set of integers between – 2 and 0, the set of whole numbers between 0 and 2, the set of multiples of 5 between 8 and 11, the set of factors of 6 between 5 and 7, 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 5 between 8 and 80, the set of factors of 3 between 5 and 73, 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'. For example, for the set of all rational numbers, the universal set can be the set of real numbers.
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