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### Onto (Surjective) Functions

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Date: 11/03/98 at 03:47:31
From: Misaki Taro
Subject: Onto (Surjective) function

According to some texts, the "onto" function is defined in the
following way:

A function f from A to B is called onto if for all b in B there is
an a in A such that f(a) = b. Such functions are also called
surjective.

I don't know why most of the definition of an onto function is defined
in an "inverted" way. Why is "for all b in B" mentioned first, while
"onto function f from A to B" is the very center of the definition? Is
it necessary to state the range first, not the domain?

To be honest, I don't quite understand this kind of function. Would
you please explain this family of functions to me in a more detailed
and illustrative way? Thanks a lot.
```

```
Date: 11/03/98 at 05:57:12
From: Doctor Allan
Subject: Re: Onto (Surjective) function

Hi Misaki,

Let me give you a specific example:

Consider the function f(x) = 3*x and the sets:

A = {2,3,9,12}
B = {6,9,27,36} and
C = {6,9,21,27,36}

The function f from A to B is onto because no matter which element you
take in B, you can find an element in A such that 3 times that element
equals the element from B.

Take as an example 27 in B. The element 9 in A equals 27 when
multiplied by 3, and it works this way for all elements in B since:

2*3 = 6, 3*3 = 9, 9*3 = 27, and 12*3 = 36

But when we let the function f go from A to C it is not onto. That's
because the element 21 in C can't be paired with one in A - there is
no element in A such that 3 times that element equals 21.

Hope this helps!

Sincerely,

- Doctor Allan, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Analysis
High School Functions

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