Cardinality
It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable. To see this, we show that there is a function f from the Cantor set C to the closed interval that is surjective (i.e. f maps from C onto ) so that the cardinality of C is no less than that of . Since C is a subset of, its cardinality is also no greater, so the two cardinalities must in fact be equal, by the Cantor–Bernstein–Schroeder theorem.
To construct this function, consider the points in the interval in terms of base 3 (or ternary) notation. Recall that some points admit more than one representation in this notation, as for example 1/3, that can be written as 0.13 but also as 0.022222...3, and 2/3, that can be written as 0.23 but also as 0.12222...3. (This alternative recurring representation of a number with a terminating numeral occurs in any positional system.) When we remove the middle third, this contains the numbers with ternary numerals of the form 0.1xxxxx...3 where xxxxx...3 is strictly between 00000...3 and 22222...3. So the numbers remaining after the first step consists of
- Numbers of the form 0.0xxxxx...3
- 1/3 = 0.13 = 0.022222...3
- 2/3 = 0.122222...3 = 0.23
- Numbers of the form 0.2xxxxx...3.
This can be summarized by saying that those numbers that admit a ternary representation such that the first digit after the decimal point is not 1 are the ones remaining after the first step.
The second step removes numbers of the form 0.01xxxx...3 and 0.21xxxx...3, and (with appropriate care for the endpoints) it can be concluded that the remaining numbers are those with a ternary numeral whose first two digits are not 1. Continuing in this way, for a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit a numeral representation consisting entirely of 0s and 2s. It is worth emphasising that numbers like 1, 1/3 = 0.13 and 7/9 = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.2222...3, 1/3 = 0.022222...3 and 7/9 = 0.2022222...3. So while a number in C may have either a terminating or a recurring ternary numeral, one of its representations will consist entirely of 0s and 2s.
It has been conjectured that all algebraic irrational numbers are normal. Since members of the Cantor set are not normal, this would imply that all members of the Cantor set are either rational or transcendental.
The function from C to is defined by taking the numeral that does consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a binary representation of a real number. In a formula,
For any number y in, its binary representation can be translated into a ternary representation of a number x in C by replacing all the 1s by 2s. With this, f(x) = y so that y is in the range of f. For instance if y = 3/5 = 0.100110011001...2, we write x = 0.200220022002...3 = 7/10. Consequently f is surjective; however, f is not injective — interestingly enough, the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed. For instance, 7/9 = 0.2022222...3 and 8/9 = 0.2200000...3 so f(7/9) = 0.101111...2 = 0.112 = f(8/9).
So there are as many points in the Cantor set as there are in, and the Cantor set is uncountable (see Cantor's diagonal argument). However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints. As noted above, one example of such a number is ¼, which can be written as 0.02020202020...3 in ternary notation.
The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. The irrational numbers have the same property, but the Cantor set has the additional property of being closed, so it is not even dense in any interval, unlike the irrational numbers which are dense in every interval.
Read more about this topic: Cantor Set, Properties