Introduction of Sets

A set is defined as a collection of distinct objects of the same type or class of objects. The purposes of a set are called elements or members of the set. An object can be numbers, alphabets, names, etc.

Examples of sets are:

1. A set of rivers of India.
2. A set of vowels.

We broadly denote a set by the capital letter A, B, C, etc. while the fundamentals of the set by small letter a, b, x, y, etc.

If A is a set, and a is one of the elements of A, then we denote it as a ∈ A. Here the symbol ∈ means -"Element of."

Sets Representation:

Sets are represented in two forms:-

a) Roster or tabular form: In this form of representation we list all the elements of the set within braces { } and separate them by commas.

Example: If A= set of all odd numbers less then 10 then in the roster from it can be expressed as A={ 1,3,5,7,9}.

b) Set Builder form: In this form of representation we list the properties fulfilled by all the elements of the set. We note as {x: x satisfies properties P}. and read as 'the set of those entire x such that each x has properties P.'

Example: If B= {2, 4, 8, 16, 32}, then the set builder representation will be: B={x: x=2ⁿ, where n ∈ N and 1≤ n ≥5}

Standard Notations:

x ∈ A	x belongs to A or x is an element of set A.
x ∉ A	x does not belong to set A.
∅	Empty Set.
U	Universal Set.
N	The set of all natural numbers.
I	The set of all integers.
I0	The set of all non- zero integers.
I+	The set of all + ve integers.
C, C0	The set of all complex, non-zero complex numbers respectively.
Q, Q0, Q+	The sets of rational, non- zero rational, +ve rational numbers respectively.
R, R0, R+	The set of real, non-zero real, +ve real number respectively.

Cardinality of a Sets:

The total number of unique elements in the set is called the cardinality of the set. The cardinality of the countably infinite set is countably infinite.

Examples:

1. Let P = {k, l, m, n}
The cardinality of the set P is 4.

2. Let A is the set of all non-negative even integers, i.e.
A = {0, 2, 4, 6, 8, 10......}.

As A is countably infinite set hence the cardinality.