Hmm. My intuition says that your A and B are “pretty much the same size”. Sure, there are infinitely many times that they switch places, but they do so about as regularly as possible and they’re always close.
If A is “numbers with an odd number of digits” and B is “numbers with an even number of digits” that intuition starts to break down, though. Not only do they switch places infinitely often, but the extent to which one exceeds the other is unbounded. Calling A and B “pretty much the same size” starts to seem untenable; it feels more like “the concept of being bigger or smaller or the same size doesn’t properly apply to the pair of A and B”. (Even though A and B are well defined, not THAT hard to imagine, and mathematicians will still say they’re the same size!)
If A is “numbers whose number of digits is a multiple of 10”, and B is all the other (positive whole) numbers, then… I start to intuitively feel like B is bigger again??? I think this is probably just my intuition not being able to pay attention to all the parts of the question at the same time, and thus substituting “are there more multiples of 10 or non-multiples”, which then works the way you said.
Hmm. My intuition says that your A and B are “pretty much the same size”. Sure, there are infinitely many times that they switch places, but they do so about as regularly as possible and they’re always close.
If A is “numbers with an odd number of digits” and B is “numbers with an even number of digits” that intuition starts to break down, though. Not only do they switch places infinitely often, but the extent to which one exceeds the other is unbounded. Calling A and B “pretty much the same size” starts to seem untenable; it feels more like “the concept of being bigger or smaller or the same size doesn’t properly apply to the pair of A and B”. (Even though A and B are well defined, not THAT hard to imagine, and mathematicians will still say they’re the same size!)
If A is “numbers whose number of digits is a multiple of 10”, and B is all the other (positive whole) numbers, then… I start to intuitively feel like B is bigger again??? I think this is probably just my intuition not being able to pay attention to all the parts of the question at the same time, and thus substituting “are there more multiples of 10 or non-multiples”, which then works the way you said.