Mathematics FUNCTION

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FUNCTION

A function is a special kind of relation

Definition-1 :
Let `A` and `B` be two sets and let there exist a rule or manner or correspondence `'f'` which associates to
each element of `A` to a unique element in `B` ,then `f` is called a function or mapping from `A` to `B`. It is
denoted by the symbol

`f : A -> A overset (f) (->) B`

which reads `f` is a function from `A` to `B` or `f` maps `A` to `B,`

`f` an element `a in A` is associated with an element `b in B` then `b` is called the `f` image of `'a'` or ' image of
a under `f'` or 'the value ofthe function fat `a'`. Also `a` is called the pre-image of `b` or argument of `b` under
the function[ We write it as

`b = f(a)` or `f : a -> b` or `f : (a,b)`

Definition-1 :
Let `A` and `B` be two sets and let there exist a rule or manner or correspondence `'f'` which associates to
each element of `A` to a unique element in `B` ,then `f` is called a function or mapping from `A` to `B`. It is
denoted by the symbol

`f : A -> A overset (f) (->) B`

which reads `f` is a function from `A` to `B` or `f` maps `A` to `B,`

`f` an element `a in A` is associated with an element `b in B` then `b` is called the `f` image of `'a'` or ' image of
a under `f'` or 'the value ofthe function fat `a'`. Also `a` is called the pre-image of `b` or argument of `b` under
the function[ We write it as

`b = f(a)` or `f : a -> b` or `f : (a,b)`

Function as a set of ordered pairs:

A function `f: A -> B` can be expressed as a set of ordered pairs in which each ordered pair is such that
its first element belongs to `A` and second element is the corresponding element of `B.`

As such a function `f: A -> B` can be considered as a set of ordered pairs `(a,f( a))` where `a in A` and
`f(a) in B` which is the `f` image of a. Hence `f` is a subset of `A xx B.`

Definition-2 :

A relation `R` from a set `A` to a set `B` is called a function if

(i) each element of `A` is associated with some element of `B.`

(ii) each element of `A` has unique image in `B.`

Thus a function `'f'` from a set `A` to a set `B` is asubsetof `A xx B` in which each `'a'` belonging to `A` appears
in one and only one ordered pair belonging to `f` Hence a function `f` is a relation from `A` to `B` satisfying
the following properties:

Every function from `A -> B` satisfies the following conditions.

(i) `f cc AxxB`

(ii) `forall a in A => (a,f(a)) in f` and

`(a,b) in f & (a,c) in f => b = c`

Thus the ordered pairs of `f` must satisfy the property that each element of `A` appears in some ordered
pair and no two ordered pairs have same first element.

Note: Every function is a relation but every relation is not necessarily a function.

A function `f: A -> B` can be expressed as a set of ordered pairs in which each ordered pair is such that
its first element belongs to `A` and second element is the corresponding element of `B.`

As such a function `f: A -> B` can be considered as a set of ordered pairs `(a,f( a))` where `a in A` and
`f(a) in B` which is the `f` image of a. Hence `f` is a subset of `A xx B.`

Definition-2 :

A relation `R` from a set `A` to a set `B` is called a function if

(i) each element of `A` is associated with some element of `B.`

(ii) each element of `A` has unique image in `B.`

Thus a function `'f'` from a set `A` to a set `B` is asubsetof `A xx B` in which each `'a'` belonging to `A` appears
in one and only one ordered pair belonging to `f` Hence a function `f` is a relation from `A` to `B` satisfying
the following properties:

Every function from `A -> B` satisfies the following conditions.

(i) `f cc AxxB`

(ii) `forall a in A => (a,f(a)) in f` and

`(a,b) in f & (a,c) in f => b = c`

Thus the ordered pairs of `f` must satisfy the property that each element of `A` appears in some ordered
pair and no two ordered pairs have same first element.

Note: Every function is a relation but every relation is not necessarily a function.

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