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Onto (Surjective) Functions


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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