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A set is a well defined collection of objects.

The collection member is called number, object or element .

We shall denote sets by capital alphabets eg. A,b, ——- &

The elements by the small alphabets e.g- a,b,c,—————- z

Some reserve letters for these sets.

N: for the set of natural numbers {1,2,3,4,5,6,……………….}

Z: for the set of integers {-3,-2,-1,0,1,2,3…………………….}

$Z^+$ : for the set of all positive integers.

Q: for the set of all rational numbers.

T- the set of all irrational number

(x:x = $\frac{m}{n}$ , m,n e z, ${n\neq0}$) (x:xe r and $x\neq0$)

$Q^+$ : for the set of all positive rational number.

R: for the set of all real numbers.

$R^+$: for the set of all positive real numbers.

C: for the set of all complete numbers.

Then NCZCQCR

TCR AND $N\neq T$

**Description of sets**:- Set is often described in the following two ways.

1) ROSTER METHOD :- a set is described by listing elements,separated by commas, within fraces { } e.g the set of vowels of English alphabets described as (a,e,i,o,u).

2) SET SUICDER METHOD:- in this method a set is described by a characterizing property

P(x) of elements x in such a case the set is described .

{x:P(x) holded} or {x/P(x) holder} is read as

The set of all x such that P(x) holds

e.g:- A= (0,1,2,3,4,8,16,………………..) is written as A = {$x^2$ : xe}

**TYPES OF SETS**

**EMPTY SET**:- if it has no elements, said empty or void or nail set & denoted by $\phi$

in roster method $\phi$ is denoted by {}

builder method = {xeR : $x^2$ = -z}

**SINGLE TON SET** a set consisting of a single element e.g {5} is a single ton set.

{x:xEN} and $x^2$ =9 } is a single ton set equal to {3}

**FINITE SET**: A finite set it is either empty set or its element can be listed or counted. By natural number (1,2,3,……….) the process of listing terminates at a certain natural number n(A) and is called ordinal number or order of a finite set

e.g. set A = (1,2,3,………………….99) Natural number less then 100

**INFINITE SET **: A set whose element cannot be listed by the natural number 1,2,3 …………… n for only natural number n is called an infinite set

Example

set of all paints in a plane.

{ex er : 0

set A = (1,2,3) set B = {a.b,c,}

Here n(A) = n(B)

**EQUAL SETS**: if every elements of set A is a number of set Band every elements of set B is a number of set A, are called equal sets

Set A = {1,2,5,6}

Set B = {5,6,2,1}

**SUB SETS**: if every elements of A is a an element of B, then A is called sub set of B and denoted by $A\leq B$

“A is sub set of B”

Thus $A\leq B$ if $ \varepsilon A \longrightarrow a \varepsilon b$

$\longrightarrow$ The symbole stands for “implies”

If A is not a sub set of B we write $A \neq B$

**SUPER SET** : If A is subset of B then B is a super set of A

Note:- 1) every set is a subset of itself $A \lte B$

2) empty set is subset of every set $\phi \lte A$

3) above 1 and 2 are called improper sub set

4) A subset A of a set B is called a proper sub set of B $A \lte B$

If AB then B is super set of A

If A=B then A≤B and $B \lte A$

5) the total number of subsets of a finite set containing n elements is 2n

A finite set A containing n elements each sub set of A have r elements

Then $0 \lte r \lte n$

element is $^n C_2$ then the total number of sub set of A

= $^n C_0$ + $^n C_1$ + $^n C_2$ ………………………….. + $^n C_n = (1 + 1)^n = 2^n$ (binomial theorem )

**INTERVAL OF SUB SETS OF R**

1) **CLOSED INTERVELS** – on real line a & b two given real numbers such that a

**2) OPEN INTERVALS – on real line, a & b two given real number such that a < b , then the set of all real number x such that a < x < b is called open interval & is denoted by (a,b) or ]a,b[**

**3) SEMI OPENED OR SEMI CLOSED INTERVAL: on real line, a & b two given real number such that a < x ≤ b or a ≤ x < b is called semi open or semi closed interval & denoted by (a,b] or [a,b) or ]a,b], [a,b[Note:- the number b-a is called the length of any of the intervals.
(a,b) [a,b] [a,b) (,b]
The interval (0, $\infty$) denoted by set R+ of all non negative real number
The interval (- $\infty$,0) denoted by set R- of all negative real number
The interval (- $\infty$,$\infty$) denoted by set R of all real numberUNIVERSAL SET:a set that contains all sets under consideration i.e. it is a super set of each of the given set, is called universal set & is denoted by U Thus a set that contains all set in a given contex is called the universal.**

e.g A= {1,2,3}, B = {2,4,5,6}, c= {1,3,5,7,}

then u = {1,2,3,4,5,6,7}

**e.g. when we are using intervals on real line, the set R of real numbers is taken as the universal set.**

**POWER SET**: the collection or family of all sub sets of a is called the power set & is denoted by p(A) = {s:s CA}

**Let set A = {1,2,3} Then the sub set are- $\phi}$, {1}, {2}, {3},{1,2}, {1,3}, {2,3}, {1,2,3} = P(A) (a set having n elements has 2n sub set s) ($2^3 = \varepsilon$ sub sets)**