F3:    Unique Solution to    x+a=b

Rosenlicht states this consequence as follows:  "For any [real numbers]  a,  b  [...] the equation  x+a=b has one and only one solution."  (see ita-2-1-7-pg17)  Rosenlicht's proof and test of solution are illustrated below with the justifications for each step explicitly given in the second column of the output matrices.

[Graphics:../Images/index_gr_35.gif]
[Graphics:../Images/index_gr_36.gif]
[Graphics:../Images/index_gr_37.gif]
[Graphics:../Images/index_gr_38.gif]

In the above cited paragraph, Rosenlicht makes the following additional points:  (1) If we let  b=a  in  F3.1,  then we get  x=0,  which shows that the neutral element for addition is unique.  (2) If we let  b=0  in  F3.1,  then we get  x=(-a),  which shows that, for any real number  a,  the additive inverse is unique.

Definition of Subtraction

Convention

ita


Converted [ in part ] by Mathematica      December 4, 2007