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A relation (“ is Religion of”) between SET A to SET B is a sub set of A×B

Where ‘R ≤ A×B

If SET A & SET B is not Vol D SET & If (a,b) $\varepsalon$ R read as a is related to b by relation R’ if (a, b) R then a ×b (a is not related to b & y any relation R

**Total Number Of Relation**: If SET A and SET B not empty finite sets consisting of m and n elements then A×B consists of mn ordered pairs and total no. of sub sets of A×B is $2^{mn}$

Among these $2^{mn}$ relations the void relation $\phi$ & the universal relation A×B are trivial relations from A to B.

**Domain And Range Of A Relation**– relation to SET A to SET B all first components or co ordinates of the ordered pairs belonging to R is called the domain while the set of all second components or co-ordinates of the ordered pair in the R is called the range of R.

**Relation On A Set**– A non raid SET- A to relation is set (SET A) = A×A is called a relation on SET A

Inverse relation:- If SET A and SET B two sets and R is relation between them then inverse of R denoted by $R^{-1}$, is a relation from B to A

$R^{-1}$ = {(b,a): (a,b) $\varepsilon$ R}

**Functions**: a relation F from SET A to SET B i.e a subset of A×B is called function L or a mapping (or a map) from A to B if

(i) For each a $\varepsilon$ A there exists bε B such that (a,b) f

(ii) (a,b) $\varepsilon$ f and (a,c) $\varepsilon$ f b=c

A non void set f of A×b is a function from A to B if each elements of A appears in some ordered pairs in f and no two ordered pairs in f have the first element.

If (a,b) $\varepsilon$ f then b is called the image of a under f.

e.g. set A= (2,3,4)

set B = (3,4,5)

f1, f2, f3 is three sub set of A×B as below.

f= {(1,2,3), (3,4), (4,5)}

f2 = {(2,3), (2,4) (3,4) (4,5)}

f3= {(2,4), (3,4)}

here f1 is a function from A to B

f2 is not a function from A to B because 2 $\varepsilon$ A has two images 3 and 4 in B.

f3 is not a function A to B because 4 $ varepsilon$ A has no image in B.

if a function F is expressed as the set of ordered pair, the domain of f is the set of all first component(elements ) of member s of f & the range of f is the set of second components of member of f.

**Function As A Correspondence**

If A and B is non empty sets, then a function f from set A to set B is a rule or method or correspondence. Which associate elements of set A to set B such that

(1) All elements of Set A are associated to elements in set B.

(2) An element of set A is associated to a unique element in set B.

If ‘ f ’is a function from a set A to set B then we write f:A $\rigtharrow$ B or A $\rightarrow$ B and read as “f is a function from A to B”.

If an elements a $\varepsilon$ A is associated to an elements bε B then B is called the f image of a’ or image of a under f’ or the value of the function f at a’

A is called the pre image of B under function f & write b = f(a)

Description of a function:- if f:A $\rightarrow$ B be a function such that the set A consists of a finite number of elements than f(x) described by listing the values it attains at different paints of its domain

**Domain Co Domain And Range Of A Function**:

IF f :A $\rightarrow$ B , then the set A is known as the domain of f and the set B is known as the co domain of f. The set of all of image of elements of A is known as the range of f or image set of A under f and is denoted by f(A)

F(A)= {f(x): x $\varepsilon$ A} = range of f or f(A) ≤ B

e.g. Set A= {-2,-1,0,1,2,} and B= {0,1,2,3,4,5,6}

Consider a rate f(x) = $x^2$

then f(-2) = $(-2)^2$ = 4

f(-1) = $(-1)^2$ = 1

f(0) = $(0)^2$ = 0

f(1) = $(1)^2$ = 1

f(2) = $(2)^2$ = 4

as above each elements of A is associated to a unique elements of B so, $f:A \rightarrow B$ is given by f(x) = $x^2$ is a function

Here domain (f) = A {-2,-1,0,1,2}

Range (f) = {0,1,4}

**EQUAL FUNCTION**

Two functions f & g are said to be equal, if

(1) Domain of f = domain of g.

(2) Codomain of f = co-domain of g

(3) F(x) = g(x) for every x beginning to their common domain then function f = function g

**Real Function** : if domain and co domain are sub set of :

(A) (B)

The set R of all real number , function are called real function.

Domain of real function:- real functions are described. By providing the general for mald for finding the image of elements in its domain.

Range Of Real Function:- the range of a real function of a real variable. Is the set of all real value fallen by f(x) at paints in its domain.

Constant Function:- if K is a fixed real number

Then f(x) = k (x $\varepsilon$ R)

The graph of a constant function f(x) = K is a straight line. Parallel of x – axis above or below of x axis according positive or negative value of K, if k = 0 the straight line is coincident of x axis