placid geyser
Im not entirley familiar with how mathematicians deal with the sizes of infinite sets so please feel free to correct any mistakes.
From what i understand, 2 sets have the same size if there exists a one to one mapping from one to the other (ie a bijective function).
I also know that there exists no said mapping between the Natural numbers (N) and the Real numbers (R) (Cantor's diagonal proof) and that there does exist one between the Real numbers and the powerset of the Natural numbers, so we can say that | N | != | R | = | P(N) |.
However i cant seem to figure out how the following mapping fails to be one to one between N and P(N):
Take a natural number and consider its binary representation (one to one). From this binary number construct a set that includes the position of each 1 and does not include the position of each 0. So for example 8 -> 1000 -> {3}, 3 -> 11 -> {0, 1}, 0 -> 0 -> empty_set
For this to work one would need to consider 0 a natural number.
The same goes the other way around, take a subset of natural numbers and for each element x in said subset sum up 2^x (ie acquire the value of the binary representation). Empty set -> 0, {1, 4} -> 2^4 + 2^1 = 18
red pond
bleak sandal