[MATHEDCC] Algebra Qestion ll
Richard Kern (canaar@igc.apc.org)
Wed, 02 Apr 1997 23:19:59 +0000
Wayne and Janey thanks for the response. I believe that you have
answered my question but I am going to try for a little more clarity in
my own mind. I am less focused on the solution to the text problem than
I am on the rigor of formally establishing Algebra as a mathematical
system. At the point where -x=4 in (Mod5,+) from the original 3-x=2, I
still believe the wheels fall off given that the only foundations thus
far presented are reflexivity, substitution, for all a,b,c elements of
Set S and operation * and group (S,*) if a=b then c*a=c*b, if a=b then
a*c=b*c, if c*a=c*b then a=b, if a*c=b*c then a=c, if a*c=b then c=a
inverse *b is a unique solution, if c*a=b then c=b*a inverse, and (a
inverse)inverse = a. I can appreciate that intuitively, addition and
subtraction are inverse operations. However the text has not explicitly
stated this axiomatically or through definition. This omission is
interesting in light of the definition of division for modular
operational systems as follows a/b=c defined as a=b times c. Further,
the text goes to some length to define the operations of addition and
subtraction separately in modular systems and does not provide further
exercises or direction investigating inverse operations. Even division
is not explicitly defined as an inverse operation of multiplication (I
suppose having to do with the problematic zero). Even given that
subtraction is an inverse operation of addition, I find no foundation in
the given axioms and theorems stated above for reaching what is an
obvious and unique solution to the problem. Wayne, if -x is a
notational device for indicating the inverse of x, i can intuitively
appreciate that (-x)inverse = (4)inverse yields the solution x=1.
However, the foundation for this "inversion property of equality?" is
missing. The unit in the text is groups with two operations with this
chapter being the introductory material prior to the introduction of 2
operation groups (in support of distributivity, fields etc.). Am I
correct in maintaining that with this problem, the authors have put the
cart before the horse, or in the sense of formal geometry, that I have
attempted a proof using a theorem not yet proven? Sorry for the lack of
elegance but what the hey, my first degree was music performance.
richard kern
napaskiak
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