Re: [其他] onto 與合成函數
※ 引述《dog2005xx (pp)》之銘言：
: *(表示合成函數)
: 題目:
: f:A->B.
: 假如f是onto if and only if
: wherever C is a set and g:B->C and
: h:B->C 是函數 such that g*f=h*f
: it follow that g=h
=> f is onto function
Let h,g be distinct functions,
where h:B->C, g:B->C, and g。f = h。f
since h,g are distinct, there exists α∈B such that g(α) ≠ h(α).
Since f is onto, then ∃x∈A, such that f(x) = α.
which implies (g。f)(x) ≠ (h。f)(x)
<= For arbitrary h,g, g。f = h。f, implies g=h
If f is not onto, then ∃y∈B such that ∀x∈A, f(x)≠y
Let g(y) = k1 ≠ h(y) = k2, where k1,k2∈C
(which g。f = h。f is still hold.)
But g≠h.
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