Actuarial Outpost Function definition in set theory.
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#1
05-20-2016, 09:05 PM
 Cessh Member SOA Join Date: Jan 2016 Posts: 99
Function definition in set theory.

This is a tricky question

Let's define a function F as f:A->B where domain A is the set {1, 2, 3} and codomain B is the set {7, 8, 9}.

So that f(1)=7, f(2)=8, f(3)=9

Now let us take the same function f and give it a domain {1, 2, 3, 4, 5, 6}. What would the output (image) of this function be?
#2
05-21-2016, 12:06 AM
 clarinetist Member Non-Actuary Join Date: Aug 2011 Studying for Rcpp, Git Posts: 6,870

The images of 4, 5, and 6 induced by f are undefined.
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#3
05-21-2016, 12:19 AM
 Cessh Member SOA Join Date: Jan 2016 Posts: 99

I alsoalso thought so. Does undefined means zero? Or empty?
#4
05-21-2016, 10:04 AM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,057

If it has a different domain, then it isn't the same function.
#5
05-21-2016, 06:37 PM
 Cessh Member SOA Join Date: Jan 2016 Posts: 99

Quote:
 Originally Posted by Academic Actuary If it has a different domain, then it isn't the same function.
But isn't a function usually symbolized as a blackbox with one input and one output (like in Wikipedia), which means you can vary the input of a function but you don't actually have direct control over the output?
and btw how do you know it's not the same function? It could be the same function. For example if I say the first domain was for positive integer numbers and in the second domain it was for negative integer numbers. And the function is still let's say x multiplied by 7. It doesn't change the function when I changed the domain, right?

Last edited by Cessh; 05-21-2016 at 06:43 PM..
#6
05-22-2016, 09:46 AM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,057

Quote:
 Originally Posted by Cessh But isn't a function usually symbolized as a blackbox with one input and one output (like in Wikipedia), which means you can vary the input of a function but you don't actually have direct control over the output? and btw how do you know it's not the same function? It could be the same function. For example if I say the first domain was for positive integer numbers and in the second domain it was for negative integer numbers. And the function is still let's say x multiplied by 7. It doesn't change the function when I changed the domain, right?
Input x 7 = Output by itself is meaningless. What if your input isn't a number?
#7
11-17-2017, 02:07 AM
 Z3ta Member SOA Join Date: Sep 2015 Posts: 361

Quote:
 Originally Posted by Cessh It doesn't change the function when I changed the domain, right?
Yes it does. A function is defined by its domain, codomain and the assignment of each element of the domain to an element of the codomain. Change any of those three things and it is a different function (that includes just throwing in another element of the codomain that isn’t mapped to).

What you are describing could be thought of as restricting the function

$
f:\mathbb{R} \to \mathbb{R}\\
\text{ }x \mapsto 7x\\
$

to subsets of the domain. You are still defining different functions even though you’re pointing out that they are all restrictions of a single familiar function.