Set theory question.

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

Thank You!

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- Dec 5th 2012, 04:49 PM #1
## If [TEX]f:A\to B[/TEX] is onto [TEX]B[/TEX], then [TEX]P(B)\leq P(A)[/TEX]

Set theory question.

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

Thank You!

- Dec 6th 2012, 03:11 AM #2

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- Dec 6th 2012, 03:45 AM #3
## Re: If [TEX]f:A\to B[/TEX] is onto [TEX]B[/TEX], then [TEX]P(B)\leq P(A)[/TEX]

- Dec 6th 2012, 04:51 AM #4

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## Re: If [TEX]f:A\to B[/TEX] is onto [TEX]B[/TEX], then [TEX]P(B)\leq P(A)[/TEX]

We have a surjection f from A to B. We need an injection g from P(b) to P(A).

Let b in P(B), we define g like this : g(b)=f^-1(b). g is an injection, proof by contradiction :

let b1 and b2 two subsets of B such that b1=/=b2 and g(b1)=g(b2)=S (S is some subset of A).

Let c in b1\b2 (c is in b1 and not in b2) let x=f^-1(c). x exists because f is surjective. we know that x is in S and f(x)=c.

But c=f(x) belongs to b2 also (because the image of an element of S is in b1 and in b2). This is a contradiction because c is in b1/b2.

so we have an injection from P(b) to P(A) this means by definition |P(b)<=|P(A)|

- Dec 6th 2012, 09:18 AM #5
## Re: If [TEX]f:A\to B[/TEX] is onto [TEX]B[/TEX], then [TEX]P(B)\leq P(A)[/TEX]

$\displaystyle f$ is an surjection from $\displaystyle A\to B.$

Let $\displaystyle \mathcal{C}=\left\{f^{-1}(\{b\}):b\in B\right\}$. Now $\displaystyle \mathcal{C}$ is a partition of the set $\displaystyle A$.

There is a choice function, (why?) on $\displaystyle \mathcal{C}$ so that $\displaystyle a_b\in f^{-1}(\{b\})$.

Define $\displaystyle g:B\to A$ by $\displaystyle b\mapsto a_b$.

Show that $\displaystyle g$ is an injection.