Would it be true to say that two infinite sets would always be equivalent since you could always match the index of their elements?
For instance, match the first element in $A$ to the first element in $B$, match the first element of $B$ with the first element of $A$, the same for the second elements, and so on?
This was my intuition but looking this up for verification showed that not all infinite sets are equivalent but did not mention why the above does not hold true, and that is my question.
On
The idea of having a "first element" is related to the idea of being "well-ordered."
You might wish to look this latter term up for more information.
As a specific example, the following two infinite sets cannot be matched up in a one-to-one and onto way (i.e., bijectively):
The integers, $\mathbb{Z}$, and the set of all subsets of the integers, $P(\mathbb{Z})$.
(See also the wikipage on Cantor's Theorem.)
No. It's not true. Taking "equivalent" and "matching indexes" as "equipotent", or "having the same cardinality" meaning "there exists a bijection" or a "a one-to-one correspondence" between the sets.
If $X$ is any set, then $\mathcal P(X)=\{A\mid A\subseteq X\}$ cannot be put in a one-to-one correspondence with $X$. In particular if $X$ is infinite, its power set, $\mathcal P(X)$, is not equivalent to $X$.
A very interesting thing in this case, is that $\Bbb N$ and $\Bbb Q$ can be put into a one-to-one correspondence, whereas $\Bbb N$ and $\Bbb R$ cannot be put into such matching.