15/03/14 06:03:04.26 1ktc1FSG.net
>>519
well-difined は、はっきりさせておきたい
ここは初学者も来るからね
wikipedia にあるように
「写像あるいは（一価の）関数 f は代入原理と呼ばれる条件 a = b → f(a) = f(b)
を満たす対応（一意対応）でなければならないから、同値類に対する写像をその代表元を用いて定義しようとする場面などでは well-defined 性が問題になる。
典型的なものが、代数学において商代数系（商群や商環、商ベクトル空間など）の演算を導入する場面に現れる。」と
つまりは、well-defined 性とは、代入原理と呼ばれる条件 a = b → f(a) = f(b)が成り立つかどうかと
英wikipedia では
Well-defined functions
All functions are well-defined binary relations: if there exist two ordered pairs in the function with the same first coordinate, then the two second coordinates must be equal.
More precisely, if (x,y) and (x,z) are elements the function f, then y=z.
Because the output assigned to x is unique in this sense, it is acceptable to use the notation f(x)=y (and/or f(x)=z) and to take advantage of the symmetric and transitive properties of equality.
Thus if f(x)=y and f(x)=z, then of course y=z.
An equivalent way of expressing the definition above is this: given two ordered pairs (a,b) and (c,d), the function f is well-defined iff whenever a=c it is the case that b=d.
The contrapositive of this statement, which is equivalent and sometimes easier to use, says that b≠d implies a≠c.
In other words, "different outputs must come from different inputs."
In group theory, the term well-defined is often used when dealing with cosets, where a function on a quotient group may be defined in terms of a coset representative.
Then