Ideas and Symbols to be Used with the Order Property

"The [real numbers  a  in  R+  (previously referred to as 'R plus')]  will of course be called positive, those such that  -a  [is  in  R+  will be called] negative.  From the [order property] we shall deduce all the usual rules for working with inequalities."  (see ita-1-2-2-pg19)

"To be able to express the consequences of [the order property] most conveniently we introduce the relations   >   and   < .   For  a, b  [in R] either of the expressions  a>b  or  b<a  (read respectively as  'a is greater than  b'  and  'b  is less than  a')  will mean that  a-b  [is in R+].  Either of the expressions  [Graphics:../Images/index_gr_100.gif]  or  [Graphics:../Images/index_gr_101.gif]  will mean that  a>b  or  a=b."  (see ita-2-2-3-pg19)

Note that the greater and less than symbols point toward the smaller number.  We may use this as a reminder by thinking that the width of the symbol indicates the size of the number:  The wide end is at the larger number.  Also:  Expressions which have real numbers separated by the greater than, less than, or equality symbols can be referred to as relations.

if ... and ... only if


Converted [ in part ] by Mathematica      December 4, 2007