FUNCTION AND RELATION

ORDERED PAIRS ,DOMAIN,RANGE, RELATION AND FUNCTION

In order to know what a function is we must first have to know what a Ordered Pair,Cartesian Product, Relation,Domain and Range is.

Ordered pairs:An ordered pair consists of two elements, Say a and b ,in which one of them ,say a, is the first element and b is the second element. An ordered pair is denoted by

                                                     (a,b)   

If we do not care about the ordered  of a and b ,say (b,a), would be ordered pairs. Here (a,b) and (b,a) can be two different ordered pairs , unless a=b.So, when a pair does carry a significance property is called a ordered pair.

 Example:We want to show the age and weight of each student in a class. we can form ordered pairs  like (a,w),in which the first element indicates the age (years)and the second element indicates the weight(in pounds).Here (a,b)=(19,127) is a ordered pair but (127,19)=(b,a) is not a ordered pair because it would be hardly fit any student anywhere.

 Cartesian product/Direct product:(Process of generation of ordered pairs)

The product of sets is called Cartesian product.If two sets ,A={1,2} and B={3,4} and we wished to form all the possible ordered pairs with the first element taken from set A and the second element from set B.Here we get 4 ordered pairs; (1,3),(1,4),(2,3),(2,4). Then the Cartesian product of sets A and B,denoted by A x B (read A cross B),is the set of 4 ordered pair. that is,

A x B ={ (1,3),(1,4),(2,3),(2,4) }      (Roster method)

A x B ={a,b} /a∈ A and b∈ B}      (Set builder method)

Relations(R): Any  subset of the Cartesian product will constitute RELATION; that is. Relation is a connection between two sets A and B such that relation is a part or a sub set of the Cartesian product A x B,e.i

R ⊆A x B

Example:R consists  all the ordered pairs beginning with an odd number where A={1,2} and B={3,4}

Here the Cartesian product consists all the possible ordered pairs; that is,

AxB={ (1,3),(1,4),(2,3),(2,4) };But Relation, must be R={ (1,3),(1,4) } as it consists the ordered pairs beginning with an odd number.

Domain: Let R be a relation from A to B ,that is , let R be a subset of A x B.The domain D of the relation R is the set of all the first elements of the ordered pairs of R;that is,

D={ a / a ∈ A,  ( a,b) ∈ R };

Example:Let A=={ (1,2,3,4 },B={ (a,b,c, } and R={ (2,a),(4,a),(4,c) }.Then the the domain  of  R is the set D={ 2,4 }

 

Range (E):The range E of the relation R consists of all the second elements which in the ordered pairs R .

Example:If R={ (2,a),(4,a),(4,c) }.Then the range is the set of R is

E={ a,c }

Function is a special  relation for which each value from the first set is associated with exactly one value from the second set e.i   each x (input) is paired with exactly one y(output).Here x has an unique association with y. That is ,each input must have an output and it provides us a unique output.If we assign x as a child and y as a mother ,then we can tell each child has only one mother;It is not possible to have two mothers of a single child. But it is possible to have two or three children of a mother.

In the diagram bellow 1(a) is a function as each and every element of set A(x=input) connected  with one element of set B(y=output). and 1(d) is also a  function . But 1(b) and 1(c) are not function as in 1(b)  element 1 has two connection with a and b and in 1(d) the element 3 has no connection with any element of set B.

Figure-1:Function and Relation

Mathematically, a function f of A into B is a subset of A x B in which each a ∈ A appears in one and only one ordered pair belonging to f. that is,

It should be mentioned that a function must be a  a relation but a relation may not be always a function.

In the function y=f(x), x is referred as the argument (independent variable)of the function, and y is called the value( dependent variable) of the function.                    

Notation of function:    y=f(x)

It is  read f of X. X is the variable we are putting into the function. Any letter can be used .For example, if we had three different functions y1,y2 and y3 we could represent them in function notation as

f ( x)=y1,

g( x)=y2

and        h(x ) =y3

GRAPHICALLY VERTICAL LINE TEST

If a graph represents a function  , a vertical line drawn on the graph will cross the function only once.For every vertical line it doesn’t matter where we draw the vertical line.

Figure-2:Which are functions

In the above figure  X represents  input values and  Y represents  output values. So this basically vertical line represents input  value.In figure 2(b) and 2(c) we don’t get a function as the vertical line crosses the graph in two points.i.e  we have two  two different Y values for a x values. Here from the graph  2(a) and 2(b) ,we get function.

Finally we can show the

Relation Function
1.Relation is a connection for  a value of the first set is associated with one  or more than one value from the second set.

2.Unique connection  of a with b where       a ∈A and b ∈ B is not  necessary for a relation that is in a relation  an independent variable(x) can be connected with more than one dependent variable.

3.A relation may not be always a function.

4. y2=x is a relation but not a function as we get  two y values for a x value.

5.Graphically  the figure bellow is a relation but not a function.

1.Function is a special  connection for which each value from the first set is associated with exactly one value from the second set.

2. Unique connection  of a with b where  a ∈A and b ∈ B is must  necessary for a function. that is in a function  an independent variable(x) is connected with only one dependent variable.

3..A function must be a  a relation.

4.y=x2 is a function as we get one y value for a x value.

5.Graphically the figure bellow is a function as well as relation  .

 

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