Re: [代數] 關於集合的證明
※ 引述《kye8546 (阿愷)》之銘言：
: A、B為兩集合
: 若A\B和B\A等勢，證明A和B等勢
: 主要是不明白A∩B的部分怎麼確定它們有對射
Let I:A∩B → A∩B be a bijective function
(I的存在性證明要去看集合論的書)
and f:A\B → B\A be a bijective function
(by the definition of equipotence of A\B and B\A)
Claim (f∪I):(A\B)∪(A∩B) → (B\A)∪(A∩B)
is a bijective function.
Just prove the injection part:
If (x,z) ∈ (f∪I) and (y,z) ∈ (f∪I),then z∈(B\A)∪(A∩B).
since (B\A)∩(A∩B)=ψ
if z is in B\A, z is not in A∩B,
then both (x,z) and (y,z) are in f
thus implies x=y (because f is injective)
on the other hand, if z is in A∩B, z is not in B\A
then both (x,z) and (y,z) are in I
thus implies x=y again.(because I is injective)
hence f∪I is injective.
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