Math Is Fun Forum

Kazy

Member

Registered: 2006-01-24

Posts: 37

Proving something is well defined

I need to prove that multiplication on Q is well defined. I have no idea really what well defined even means. From what I understand from reading, it means that multiplication on Q does not depend on the choices of integers to represent the equivalence classes. However, I have no idea how to go about proving that. Anyone have any ideas?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Proving something is well defined

English version:
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Think of well defined as defined well.  When is something defined well?  When you know precisely what it is by the definition!

So the definition we have is that a * b = c, where a, b, c ∈ Q.  If we know exactly what a and b are, do we know exactly what c is?  Or is it a little fuzzy?

Here's any example: y = x².  If I tell you what y is, can you tell me what x is?  Well, no, not exactly.  If I say y is 9, x could be 3 or it could be -3.  See what I mean by fuzzy?  It's not defined well, you don't know exactly what x is.

On the contrary, if I tell you y = x, and then tell you what y is, you could tell me what x is every time.  That's pretty well defined, right?
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Math (words) version:
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Well defined simply just means that if two elements of a set are eqaul, and there is a function that maps these elements to another set, then their mapped versions will also be equal.
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Math (symbol) version:
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A map f is well defined when f:A⇒B, for all a1,a2∈A if a1 = a2 then f(a1) = f(a2)
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I did that mostly to show that mathimaticans have such a great language that something which takes paragraphs in english can be shortned down to one sentence in pure mathish.

What you need to show is that if a = b and c = d, then a * c = b * d.  Try it, and see what you can come up with.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."