Functions (Domain, Co-domain and Range)

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FUNCTIONS;


1. A ( Domain)   (set of Children)                      
2. B (Co-Domain) (set of Women)

1.A and B are non-empty sets

make relation
Relation: f (mother)

Let us understand what is function,
this relation is a perfect example of function.

"Every child has a Mother and only one Mother  i.e. No child is without Mother and Every child  has a unique mother".

2(a).Every element of set A has a unique image in set B.

f is a relation which associates each element of set A with unique element of set B, then 'f' is called a function from A to B.

Domain A = \{ a,b,c\} , Co-Domain B = \{ 1,2,3,4,5\}

"No child is without mother"
"No child has two mothers"

2(b).if any element of set A has no image i.e. it is not associated with any element of set B, it is not a function and also if any element of set A has many images, it is not a function



"It is possible that a woman has no child"

3.it is possible for a function 'f' that element of set B has no pre-image i.e. it is not associated with any element of set A

here elements 3 and 4 of set B have no pre-image.
"a woman may have more than one child"

4.it is possible for a function that element of set B has many pre-images (not unique) i.e. it is associated with many elements of set A


here 'e' has two pre-images 'b' and 'c'

"Set of all the women called Co-Domain"
and "all the women who have at least one child called RANGE"

5.The set of the elements of B, which are the images of elements of set A is called Range of f
OR
 the set of the elements of B, which have pre-images in set A is called Range of f.


Range = \{ 1,2,5\}                                                 Range = \{ d,e\}
co - domain,B = \{ 1,2,3,4,5\}                           co - domain,B = \{ d,e,f\}
domain,A = \{ a,b,c\}                                           domain,A = \{ a,b,c\}



if a is the element of set A i.e.a \in A then the element of B which is associated to a is called image of a and denoted by f(a)

 We should always remember that the Range is a Subset of Co-Domain.

A function f is a rule that assigns to each element x in a  A  set exactly one element called  f(x)   B

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