Your 1-1 relationship makes sense intuitively with a finite set but it breaks down with the mathematical concept of infinity. Here’s a good article explaining it, but DreamButt’s point of every set of countable infinite sets are equal holds true because you can map them. Take a set of all positive integers and a set of all positive, even integers. At first glance it seems like the second set is half as big right? But you can map them like this:
Set 1 | Set 2
1|2
2|4
3|6
4|8
5|10
6|12
If you added the numbers up on the two sets you would get 21 and 42 respectively. Set 2 isn’t bigger, the numbers just increased twice as fast because we had half as many to count. When you continue the series infinitely they’re the same size. The same applies for $1 vs $100 bills.
$1|$100
$2|$200
$3|$300
In this case the $1 bills are every integer while the $100 bills is the set of all 100’s instead of all even integers, but the same rule applies. Set two is increasing 100x faster but that’s because they’re skipping all the numbers in between.