# How many surjective function are there in infinite sets?

Say we have two sets $|A| = a$ and $|B| = b$, where $a \geq b$ and $a$ is infinite. How would you go about proving that the number of surjective functions from A onto B is $b^a$?

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HINT: Partition $A$ into sets $A_0$ and $A_1$, where $|A_0|=b$ and $|A_1|=a$. Let $f:A_0\to B$ be any bijection. For each function $g:A_1\to B$, the function $f\cup g$ is a surjection from $A$ to $B$. Use this to show that there are at least $b^a$ surjections from $A$ to $B$. Showing that there are no more than $b^a$ surjections from $A$ to $B$ is straightforward.
@aslkdnqr: You’re welcome. Just to make sure: a function is a set of ordered pairs, so $f\cup g$ just acts like $f$ on $A_0$ and like $g$ on $A_1$. –  Brian M. Scott Oct 18 '13 at 15:57