1. ## Cardinals - Exponents

I have a problem with cardinals:

k,j,l are cardinals. 0<k, k=< l

So this, I believe, should be the way to solution:

I need to prove that j^k<=j^l

so, let |a|=j, |b|=k, |c|=l

I need to prove that |a^c| >= |a^b|

( >= means larger or equal)

Now, I understand that I need to create a function from a^b to a^c and show that it is a 1-1 function.

I now that I can say that there is a function $f:b(to)c$ that is a 1-1 function (because i know that b<=c ), but I just can't seem to find a way to use it...

Thank you!~!

2. If $a$ is empty then both $a^b$ and $a^c$ are empty and the assertion holds.
Let $a$ be nonempty and let $s \in a$. Let $g$ be an injective map from $b$ to $c$ and let $c_1=Range(g)$. For every function $f:b \rightarrow a$ we define function $\tilde{f}:c \rightarrow a$ this way:

$\tilde{f}(t) = f(g^{-1}(t))$ if $t \in c_1$
$\tilde{f}(t) = s$ if $t \in c \smallsetminus c_1$

Then mapping $F$ defined as $F(f)=\tilde{f}$ is injective and maps $a^b$ to $a^c$.

3. I see, but there's one thing I didn't understand - if $c_1=Range(g)
$
, then what does it mean? That c is this group? :

{ k $| k \in g(b), b \in B$}

4. not quite, Range(g) = {k| (ex. d in b) k = g(d)}

5. It might not be precisely what I wrote, but that's how I understood it.

Is that similar to f[B], or Im(B) ?

(these are the signs that I'm familiar with)

6. Let f:A -> B be a fuction, by Range(f) (also Rng(f) or Ran(f)) I mean the same as your f[A] -whose advantage is btw. that you can also use it to write f[C], where C is a subset of A, to denote image of the set C under the map f.
Also Im(f) means the image of entire domain A under the mapping f.
In wiki they say: "Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f." ...and codomain means the whole B, which I didn't mean by Range(f). So I should probably stop using Range(f) and start using Im(f)!!