Re: The naive test for equality

Hi,

"Paul" <paul_at_test.com> wrote in message news:42ef9948$0$24030$ed2619ec_at_ptn-nntp-reader01.plus.net...
> Marshall Spight wrote:
>> Sure. Specifically, it's an equivalence relation. Let's distinguish
>> between the equality relation specifically and equivalence relations
>> in general. Equality is a much simpler thing.
>
> Is it, though? We think about 1/2 = 2/4 fine even though they have
> different representations. Maybe you mean "identity", often shown using
> a variant of the equals sign with three lines instead of two?

The standard construction of rationals, as introduced in high school, is:

Let ZxZ' be a set of all ordered pairs of integers (x,y) where x is not a zero (Z' = {Z minus 0}). Let's define an equivalence relation E as

(x1,y1) E (x2, y2) iff x1*y2 = y1*x2

Then rationals are a set Q of equivalence classes defined by the above relation. Technically, one has to *prove* that E is indeed an equivalence relations and that operations like addition and multiplication are well defined and obey the usual laws, etc.

There is no neeed to talk about some vague representations and such, one can simply speak in clear terms of integers and equivalence classes instead. Received on Tue Aug 02 2005 - 17:22:53 CDT