I have two questions about set theory and cardinals.
I know that $k_1, k_2, m$ are cardinals. I also know that $k_1 \leq k_2$. I need to prove that $(k_1)^m \leq (k_2)^m$.
I understand from what given that there is injective function between the sets of $k_1$ and $k_2$. I also know that if $|A| = |C|$, $|B|=|D|$ then $|B^A| = |D^C|$. so basically what I need to prove is that $k_2 \leq k_1$, and then with Cantor–Bernstein–Schroeder theorem say that if $k_1 \leq k_2$ and $k_2 \leq k_1$ then $k_1 = k_2$, am I correct? I don't know how to continue from here.
the second question is that I need to prove that the set of relations on the set $N$ is $c$. I understand from the question that I need to prove that $|P(N \times N)|=c$, but I'm not sure and I don't know how to continue from here.
I'm just tired from all this discrete math, maybe anyone can recommend a good resource to read so I will understand it better.
Thank in advance.