Im trying to understand this proof by Cantor.(adsbygoogle = window.adsbygoogle || []).push({});

For every set [itex] X, |X|<|P(x)| [/itex]

Proof. Let f be a function from X into P(x)

the set [itex] Y=(x \in X: x \notin f(x) ) [/itex]

is not in the range of f:

if [itex] z \in X [/itex] where such that f(z)=Y, then [itex] z \in Y [/itex]

if and only if [itex] z \notin Y [/itex], a contradiction. Thus f is not

a function of X onto P(x).

Hence |P(x)|≠|X|, the function

f(x)={x} is a one-to-one function of X into P(x) and so

|X|≤|P(x)|. it follows that

|X|<|P(x)|.

I dont understand why z cant be in Y and f(z).

I guess thats because they defined it that way.

Is it because we want to find a one-to-one function from

the set to the power set, and because we want it to be one-to-one

we want to map every x to a unique element in the power set we don't want x to get mapped to itself. Is that the reason. And do we need z to be in Y and f(z) for it to be onto.

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# Question about the powerset.

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