**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.**I**f 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):**** A**ny 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 }

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**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 y_{1},y_{2} and y_{3} we could represent them in function notation as

f ( x)=y_{1},

g( x)=y_{2}

and h(x ) =y_{3}

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. y 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=x 5.Graphically the figure bellow is a function as well as relation . |

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