Actuarial Outpost Left / Right Inverse
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#1
05-15-2016, 06:38 AM
 Cessh Member SOA Join Date: Jan 2016 Posts: 99
Left / Right Inverse

I know this probably is not something that will be asked in the exam. But just out of pure curiosity..

if I have the function F and G. So that G is the left inverse of F. Because G o F = Idx.

My question is why is it named "left" inverse if it is like that? Is it because of the position of G in the composition in relation to F (G is at the left of F in the G o F = Idx equation)? Or is it named "left" inverse because of some other reason?

And of course vice versa for "right" inverse?

Regards.

Last edited by Cessh; 05-15-2016 at 06:51 AM..
#2
05-15-2016, 07:39 AM
 whoanonstop Member Non-Actuary Join Date: Aug 2013 Location: Los Angeles, CA Studying for Spark / Scala College: College of William and Mary Favorite beer: Orange Juice Posts: 5,768 Blog Entries: 1

Quote:
 Originally Posted by Cessh I know this probably is not something that will be asked in the exam. But just out of pure curiosity.. if I have the function F and G. So that G is the left inverse of F. Because G o F = Idx. My question is why is it named "left" inverse if it is like that? Is it because of the position of G in the composition in relation to F (G is at the left of F in the G o F = Idx equation)? Or is it named "left" inverse because of some other reason? And of course vice versa for "right" inverse? Regards.
Well, at high level, yes. G is left inverse of F and F is right inverse of G. However, in this situation if we rearranged to F o G and that was Idx, then we wouldn't refer to F or G with a specific right or left inverse, but just an inverse.

I believe Clarinetist will have a more complete answer for you on this but if you were to think of matrices instead, left/right inverses are connected to the row/column ranks of the matrix. You'll have left when R > C and right when R < C.

As you've pointed out though, this is not important for exam P.

-Riley
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#3
05-15-2016, 09:36 AM
 clarinetist Member Non-Actuary Join Date: Aug 2011 Studying for Rcpp, Git Posts: 6,853

Quote:
 Originally Posted by Cessh Is it because of the position of G in the composition in relation to F (G is at the left of F in the G o F = Idx equation)?
This is how I think of it. A lot of the study of etymology of mathematical terms tends to be thrown out to focus solely on the mathematics, in my experience. Maybe someone else would know.

As Riley and you have mentioned, this is irrelevant for P. However, left and right inverses are particularly important as vocabulary for functions, especially in intro real analysis and abstract algebra. See Injection iff Left Inverse and Surjection iff Right Inverse. I'm not sure if left and right inverses are useful elsewhere.
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#4
05-15-2016, 10:14 AM
 Cessh Member SOA Join Date: Jan 2016 Posts: 99

Another possible guess :

it's called a "left " inverse because to do G o F = Idx means that it goes from the domain to the codomain, then back to the domain again (the domain is usually illustrated at the left)

It's called a "right" inverse because to do F o H = Idy means that it goes from the codomain to the domain, then back to the codomain again (the codomain is usually illustrated at the right side).

Does somebody knows thethe correct answer ?
#5
05-15-2016, 02:36 PM
 clarinetist Member Non-Actuary Join Date: Aug 2011 Studying for Rcpp, Git Posts: 6,853

Quote:
 Originally Posted by Cessh Another possible guess : it's called a "left " inverse because to do G o F = Idx means that it goes from the domain to the codomain, then back to the domain again (the domain is usually illustrated at the left) It's called a "right" inverse because to do F o H = Idy means that it goes from the codomain to the domain, then back to the codomain again (the codomain is usually illustrated at the right side). Does somebody knows thethe correct answer ?
You're probably not going to find an answer here. If you're really desperate for an answer, you could always ask on the History of Science and Mathematics StackExchange.
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#6
11-17-2017, 04:04 PM
 Z3ta Member SOA Join Date: Sep 2015 Posts: 361

The term "left-inverse" is used in abstract algebra for anything you apply on the left of a binary operation to get the identity element. This general definition is referred to here. Hence your first guess for the reason is correct. In this case function composition is the binary operation.

Some people prefer the term "retraction" for left-inverse and "section" for right-inverse. I would guess this is because a function only has a section if it is surjective which means it splits itself into sections corresponding to each element it maps to (the domain is partitioned into groups where each group comprises things which map to the same thing in the codomain). This means a section would pick out one element from each group. Similarly a function only has a retraction if it is an injection so everything in the domain can return to itself (retract) through a map going the other way (the retraction).
#7
11-17-2017, 04:31 PM
 Colymbosathon ecplecticos Member Join Date: Dec 2003 Posts: 5,805

Z3ta gives a pretty good explanation.

Here's one that might be more basic. Consider integration and differentiation, the Fundamental Theorem of Calculus says, effectively, that these are inverses of each other, but that is only partially true.

If you take the integral first, and then differentiate, you get back (exactly) your original function, however, if you differentiate first and then integrate you get back your original function plus an unspecified constant.

So, differentiation is a left inverse, but not a right inverse of integration.
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