Identity and Inverse Properties for Zero

Date: 01/07/2004 at 10:51:00
From: Sean
Subject: Properties of identity and inverse not true for subtraction

If we subtract 0 from a number and get the same number, doesn't that
make 0 an identity for subtraction?  Also, can't a number be its own 
inverse for subtraction?  I have a text book which doesn't explain
this--it simply gives the answer.  I feel subtraction should have the
identity properties in the case of all number systems which include 0.

Date: 01/07/2004 at 12:15:19
From: Doctor Peterson
Subject: Re: Properties of identity and inverse not true for subtraction

Hi, Sean.

If your text defines these properties carefully, the definitions
should be something like this:

  an element e is called an identity for the operation o if

    e o x = x  and  x o e = x  for all x in the set

  an element y is called the inverse of x if

    x o y = e  and  y o x = e

Note the two-sidedness of the definitions.

Let's apply this definition of the identity to addition:

  If 0 is an identity element, then

    0 + x = x  and  x + 0 = x  for all x; this is clearly true

Now try applying the definition of the identity to subtraction:

  If 0 is an identity element, then

    0 - x = x  and  x - 0 = x  for all x; the first is false

Since there is no true (double-sided) identity, we can't define an 
inverse.

The lack of associativity also makes it very difficult to apply the 
identity in the case of subtraction. 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/