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Well-defined
By Amars Birdi on Thursday, January 24, 2002 - 06:50 am:

I have answered quite a few questions on the well-definedness of, say, a function. I just wanted some kind of a concrete definition of what the term 'well-defined' means (this term crops up quite a bit on past Part 1A Cambridge Tripos questions).
Thanks

By Kerwin Hui on Thursday, January 24, 2002 - 10:23 am:

Amars,

A function is well-defined if it makes sense. For example, if I define a map of rationals to rationals by

f(a/b)=a2/b, a/b in Q

This will not make any sense since we will have f(1/2)=1/4 and f(2/4)=1. Another example: if G is a group, and H is a normal subgroup (i.e. gH=Hg for all g in G), then we can define a group operation on G/H by

(gH)(kH)=(gk)H

and this is well-defined: If aH=bH and cH=dH, we have b-1a and d-1c both in H. Now we have to show (bd)-1(ac) is in H, which can be proven easily (you may like to try this out). Formally, a function f(x) is well-defined if it does not depend on the way we represent x.

Kerwin

[Editor: The fundamental difficulty is that 'well-defined' is not well-defined.]