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CHAPTER II [:] The Real Number System [...] 1. THE FIELD PROPERTIES. "end quote
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F3 AND F4:   UNIQUE SOLUTIONS TO     x + a = b     AND     xa = b

Rosenlicht uses the usual convention that   xa   means   x   times   a.   The table immediately below shows a summary of his results. In order to combine the proofs into a single proof, I have excluded in   F3   the case where   a   is zero. This case can be done separately at the end.

For any real number   b   and any nonzero real number   a
F3   if :  
  x+a=b  
  then :     x   is the unique real number   b-a  
  0   is unique (from x+a=a)  
  and  
  -a   is unique (from x+a=0)
F4   if :  
  xa=b  
  then :     x   is the unique real number   b/a  
  1 is unique (from xa=a)  
  and  
  a is unique (from xa=1)  

Also:
Define:
  b-a  
  to be  
  b + (-a)  
  This is the definition of subtraction.  
Define:
  b/a  
  to be  
  ba  
  This is the definition of division.  

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I would like to use the symbol     for multiplication.  Therefore, the second equation above will be   xa = b.   I want to do this because the two proofs are the same if we have a symbol that can stand for either addition or multiplication, and for that purpose I would like to combine the two symbols inside a circle:   .   Here is a larger version that makes it more obvious what is intended:


When this symbol is encountered, it can be read either as "plus" or as "times".   Then the single proof below gives the two proofs with these two readings of the symbol.   However, the economy of having one proof is not my main reason for doing this.   Instead, I want to emphasize something about the neutral elements and inverses:   The numbers   0   and   1   are very familiar, but at this point in the logical development they are just symbols for the neutral elements which we have postulated. A similar remark, applies to   -a   and   a. They are just symbols for the additive and multiplicative inverses, and this means they should give us no discomfort:   We will develop negative numbers and fractions and powers later.   Using the combined "plus times" symbol forces us to introduce combined symbols for the neutral elements and inverses. I will use   Neu   and   Inv(a)   as shown below.   Remember:   0   and   1   are the only numbers we have so far, and   -a   and   a   are just symbols for the additive and multiplicative inverses.

Neu =
  0     if :     = +  
  1     if :     =  
           ,            Inv(a) =
  -a     if :     = +  
  a     if :     =  

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The proof uses the fundamental argument form of mathematical logic that is called modus ponens.   According to the Oxford English Dictionary, this is "The rule of logic which states that if a conditional statement if p then q is accepted, and the antecedent p holds, then the consequent q may be inferred".   Using   R   to represent the set of real numbers, we show that

if            xR   and   xa = b            then            x = bInv(a)

xa
  =  
b
Given with xR
(xa) Inv(a)
  =     b Inv(a)     Existence of inverses and nature of   +   and as functions  
  x ( aInv(a) )     =  
no change
Associativity
x Neu
  =  
no change
Existence of inverses and neutral elements
x
  =  
no change
Property of neutral elements

And the conditional is proven.

Next, we verify that the real number   b Inv(a)   is a solution of the equation   xa = b.

x a
Given
  ( bInv(a) ) a  
Solution to be tested
  b ( Inv(a)a )  
Associativity
b Neu
Existence of inverses
b
  Existence of neutral elements  

And this completes the proof.

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Exercise: Rewrite twice the expressions in the two tables immediately above. First, replace     by   +   and replace   Neu   and   Inv(a)   by their appropriate values.   Second, replace     by     and again replace   Neu   and   Inv(a)   by their appropriate values. Note that you may need to adjust the use of parenthesis. For example,   bInv(a)   can become   b + -a   which is better written as   b + (-a).

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