## 1. Sets

### Introduction

SETS

**Set:** A collection of well defined objects is called a set.

- Objects of collection is called "**element**" or "**member**".

**Properties: **

(i) All elements of a set should have special property.

(ii) They all should be differ to each other or they do not repeat.

(iii) They should be well defined.

- Name of any set is writen in capital letter of English Alphabets

A, B, C, D ........................ X, Y, Z

- Elements and members of a set is writen in small letter of English Alphabets or using number system within a brace (medium bracket) with using commas { }

Example : A = {a, b, c, d} Or B = {1, 2, 3, 4}

**Standard symbols of some special sets:**

N : the set of all natural numbers

Z : the set of all integers

Q : the set of all rational numbers

R : the set of real numbers

Z^{+} : the set of positive integers

Q^{+} : the set of positive rational numbers, and

R^{+} : the set of positive real numbers.

**Symbols and their meaning: **

**(i) " ∈ " (epsilon) : "belong to" **

If a is an element of a set A, we say that “ a belongs to A”

in symbolic form we write it as : a ∈ A

**(ii) " ∉ " : "not belong to" **

When any element which not belong to any given set then we use symbol ∉ : "not belong to"

if a is not an element of set A. we say that "a not belong to A"

in symbolic form we write it as : a ∉ A

**(iii) " ⇒ " : emplies **

In common language "**⇒**" means "emply toward ..... also ......"

**(iv) " = " : Equal to **

A = B, it means set A is equal to set B.

**(v) " ≠ " : Not equal to **

A ≠ B, It means set A not equal to set B.

**(vi) “⇔” **is a symbol for two way implications, and is usually read as " if and only if ".

(vii) " : " : colon Here in the set colon stands for "such that"

**Representation of sets: **

Every set is introduced by its elements. So for expressing a set before we have to express its elements.

Now, Method to express the elements of a set.

**(1) Roster Or tabular form : **

In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }. For example,

(i) First five natural numbers:

A = {1, 2, 3, 4, 5}.

(ii) First four letter of English Alphabets

B = {a, b, c, d}.

(iii) The set of all natural numbers which divide 42

C = {1, 2, 3, 6, 7, 14, 21, 42}.

(iv) The set of all vowels in the English alphabet.

D = {a, e, i, o, u}.

**Note: **

In Roster Form no-element is generally repeated

Example:

The set of letters forming the word 'ELEMENT'

E = {E, L, M, N, T}

Here no-element has been repeated.

**(2) Set-builder form **

In set-bulder form, all the elements of a set possess a single common property

which is not possessed by any element outside the set.

Example:

D = { a, e, i, o, u}

All elements of set A have a common property e.i vowel in English alphabet.

Then, in set-builder form:

D = { x : x is a vowel in english alphabet }

This set will be read as :

" the set of all x such that x is a vowel in english alphabet. "

C = {y : y is a natural number which divides 42}

B = (z : z is a first four letter of english alphabet}

Other examples

(i) A = {1, 2, 9, 25} Roster form

Set-builder form

A = {x : x = n^{2}, where n **∈ **N and n < 6}

(ii) B = {P, R, I, N, C, A, L}

Set-builder form

B = {x : x is a letter of the word PRINCIPAL}

(iii) C = {1, 2, 3, 6, 9, 18}

Set-builder form

C = { x : x is a positive integer and is a divisor of 18}

(iv) D = (3, 6, 9, 12}

Set-builder form

D = {x : x = 3n where n **∈ **N and 0 < n < 5}

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