Re: The naive test for equality

"VC" <boston103_at_hotmail.com> wrote in message news:jpGdnbccFZCrUGvfRVn-gQ_at_comcast.com...

> Not quite right. In the case of rationals equality, you treat the
> equivalence class, as a whole, as a single element. E.g, for integers
> you'd say 2=2; for rationals you'd say [5/10] = [1/2], no difference
> really since both [5/10] and [1/2] is the *same* element. In other words,
> your *equality* relation pair would be, say, for integers (1,1) and for
> rationals (E_half, E_half), where E_half = {1/2, 2/4,, 5/10, ..} etc.

Right. The entire equivalence class is a single element as viewed by the rationals engine.
In order to manipulate this "single element" as data, we need a symbol for it, to represent it.

So we choose one of the elements of the original set to stand as a representative of the entire set that is going to be seen as an element. In this case we might choose the rational with the lowest denominater, namely 1/2.

Now, whenever we are given an unnormalized rational, such as 5/10, we ask the rationals engine to normalize it for us. The rationals engine knows the rule for normalizing, namely remove common factors in the numerator and denominator. So it returns 1/2, the normalized equivalent of 5/10.

If we ask the rationals engine to normalize 1/2, it will give us back 1/2.

So the process of normalizing is choosing one, out of an equivalence class, according to some criterion, and using the symbol that represents the chosen element to act as the normalized form for the entire class. Received on Mon Aug 08 2005 - 12:37:54 CEST