Originally Posted by

**Deadstar** Right so, I have this major block with set exponentiation and unfortunately we have to do it in my set theory course with cardinals.

I'm gonna post the question and the answer and post what (after many months of thinking about it) is the best I can do on my own...

**Question**. If κ, λ and μ are cardinals with 0 < κ ≤ λ. Show that $\displaystyle \mu^{\kappa} \leq \mu^{\lambda}.$

**Solution**. Let κ = *P*(*A*), λ = *P*(*B*) and μ = *P*(*C*). Let *f*:*A*→ *B* be an injection.

Let *k*:*B*→ *A* be the surjective map defined in the proof of Prop 1.3.1 which exists because κ > 0.

Then we define a map $\displaystyle C^A \to C^B$ by G(g)(b) = g(k(b)) **(+++)**.

We claim *G* is injective.

If *G*(*g*)=*G*(*g*′) then *g*(*k*(*b*))=*g*′(*k*(*b*)).

Since *k* is surjective *k*(*b*) ranges over all possible elements of *A* and so *g*(*a*)=*g*′(*a*) for all *a* ∈ *A*. Hence, *g*=*g*′

Notice the **(+++)**, this is the part that scrambles my brain, making injections, surjections and bijections.

I'm not quite sure what $\displaystyle g$ is either, I seem to think it's a possible injection that takes an element in $\displaystyle C^A$ to $\displaystyle C^B$.

But, my attempted answer is slightly different and I want to know if it looks mildly right and if not, can you explain what the map in (+++) actually means!

So we have $\displaystyle f: A \to B$ is an injective map.

Define $\displaystyle k: B \to A$ as a surjective map from B to A.

So, for every $\displaystyle a \in A$ there is a $\displaystyle b \in B$ such that $\displaystyle a = k(b)$.

So, now the (Headbang) bit (for me)... We define a map from $\displaystyle C^A \to C^B$ by...

$\displaystyle G: g(a) = g(k(b))$...

I think I'll just stop there though, the rest of the Q is a bit easier, just the defining the map part that gets me.

Having just re-read it I'm pretty sure mine isn't even a map, it's just a renaming of $\displaystyle a$.