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# 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:

- A set of rivers of India.
- 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=2^{n}, 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. |

I_{0} | The set of all non- zero integers. |

I_{+} | The set of all + ve integers. |

C, C_{0} | The set of all complex, non-zero complex numbers respectively. |

Q, Q_{0}, Q_{+} | The sets of rational, non- zero rational, +ve rational numbers respectively. |

R, R_{0}, 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.