Proof of Addition Reversibility

  • I
  • Thread starter FAS1998
  • Start date
  • #1
43
1

Main Question or Discussion Point

How can you prove that

##f(x)=g(x) \Leftrightarrow f(x)+C=g(x)+C##
 

Answers and Replies

  • #2
13,067
9,835
I can't as long as I don't know where your elements are taken from.
 
  • Like
Likes FAS1998
  • #3
43
1
I can't as long as I don't know where your elements are taken from.
What do you mean by "elements"?
 
  • #4
13,067
9,835
What do you mean by "elements"?
##f(x),g(x),C##

If they were, as usual, from ##\mathbb{R}##, then the answer would be: because ##(\mathbb{R},+)## is a group. But if you had defined addition differently on some set, then there is not enough information about it.

E.g. ##1+1=2## and ##1+1=0## are both true, just not in the same set.
 
  • Like
Likes FAS1998
  • #5
43
1
##f(x),g(x),C##

If they were, as usual, from ##\mathbb{R}##, then the answer would be: because ##(\mathbb{R},+)## is a group. But if you had defined addition differently on some set, then there is not enough information about it.
This is what I meant.

I just looked over the wikipedia page for groups and now understand why ##(\mathbb{R},+)## is a group, but why does belonging to a group imply reversibility?
 
  • #6
jbriggs444
Science Advisor
Homework Helper
2019 Award
8,759
3,522
I just looked over the wikipedia page for groups and now understand why ##(\mathbb{R},+)## is a group, but why does belonging to a group imply reversibility?
Being in a group means existence of an additive inverse, -C.
So given that ##f(x)+C=g(x)+C## you can write down ##f(x)+C+ -C=g(x)+C+ -C##.

Given associativity, the definition of an additive inverse and the definition of zero, it is all downhill from there.
 
  • Like
Likes FAS1998
  • #7
43
1
How would we prove the reversibility of other operations such as exponentiation (for values >= 0), that don't belong to groups?
 
  • #8
13,067
9,835
How would we prove the reversibility of other operations such as exponentiation (for values >= 0), that don't belong to groups?
##x \longmapsto e^x## or ##x \longmapsto x^n## e.g. by the theorem of invertible functions or in general step by step. They aren't operations anymore, just other functions.
 
  • Like
Likes FAS1998

Related Threads on Proof of Addition Reversibility

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
23
Views
3K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
5
Views
743
  • Last Post
Replies
7
Views
5K
  • Last Post
Replies
6
Views
1K
Replies
14
Views
1K
  • Last Post
Replies
21
Views
3K
  • Last Post
Replies
4
Views
3K
Top