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I need some hints for proving that if $f:A\to B$ is onto $B$, then $P(B)\leq P(A)$. And $|B|\leq |A|$ under axiom of choice.

Thank you!

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How constructing one-to-one function $F$, where $F:P(B)\to P(A)$? – 17SI.34SA Dec 6 '12 at 3:48
up vote 3 down vote accepted

Define $\hat f\colon P(B)\to P(A)$ by setting $\hat f(X)=\{a\in A\mid f(a)\in X\}$. You can show this is an injection.

If the axiom of choice holds simply choose from $\hat f(\{b\})$ to construct the injection from $B$ to $A$.

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First, try relating the cardinality of $A$ and $B$ by using the information you have. Then, use that to generalize up to the cardinality of the power sets.

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If A and B are finite sets, I can say that |B|<|A|. But what in the case they aren't? – 17SI.34SA Dec 6 '12 at 3:16
Jebruho, relating the cardinalities requires choice. – Asaf Karagila Dec 6 '12 at 8:18

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