Is There a Bijection Between a Set and Its Power Set?

[Hurkyl]

No question, or deep answers to be found here! I just wanted to share with you a pretty formulation of Cantor's diagonal argument that there is no bijection between a set X and its power set P(X). (the power set is the set of all subsets of X)


It's based on the idea of a characteristic function: a function whose values are only 0 and 1. If Y is a subset of X, then I can describe Y by this function:

$$\chi_Y(p) := \left\{ \begin{array}{ll} 0 \quad & p \notin Y \\ 1 \quad & p \in Y \end{array}$$

So that the statement "p is in Y" is equivalent to the equation "χ_Y(p) = 1".


With a little thought, you should be able to convince yourself that there is a one-to-one correspondence between subsets of X and characteristic functions on X.

So, the question "Is there a bijection between X and P(X), the set of all subsets of X?" is equivalent to the question "Is there a bijection between X and the set of all characteristic functions on X?"


Once you get to this point, the proof becomes much cleaner.


We now assume that such a bijection exists, and we call it f. So, if x is an element of X, then f(x) is a characteristic function on X.

If you haven't seen them before, function valued functions can be a little tricky to understand... for this proof, the only thing you need to know is that we can write f(x)(y), with both x and y in X, and f(x)(y) will either be a 0 or a 1.


The next step is to write a function that represents the "diagonal": we define the function g by:

g(x) := f(x)(x)

And the key step is that we can now construct the function h:

h(x) := 1 - g(x) = 1 - f(x)(x)


The values of h are either 0 or 1, so it is a characteristic function on X. Now, since f is an invertible function (it's a bijection!), we can make this computation:

$$h(f^{-1}(h)) = 1 - f(f^{-1}(h))(f^{-1}(h)) = 1 - h(f^{-1}(h))$$

We can write f^-1(h) because f is invertible, and the range of f is the set of all characteristic functions on X, and h is one of those things.

The first step in the calculation is simply the definition of h. The second step uses the definition of invertibility, and will probably take a bit of thought to work through for those not comfortable with function valued functions. The key fact is this:

$$f(f^{-1}(h)) = h$$

so if we apply both functions to an element of X, we have:

$$f(f^{-1}(h))(x) = h(x)$$

Anyways, going back to the computation:

$$h(f^{-1}(h)) = 1 - f(f^{-1}(h))(f^{-1}(h)) = 1 - h(f^{-1}(h))$$

This is a contradiction, because the far left and the far right can never be equal. (Remember that the values of h can only be a 0 or a 1, and this would require it to take 1/2 as a value)


So, our initial assumption that there exists a bijection, f, from X to the set of all characteristic functions must have been incorrect, because we derived a contradiction. So, by the initial remarks, we have thus proven:

There does not exist a bijection from any set X to P(X), the set of all subsets of X.

 

[Zurtex]

That's very pretty in a way

Bit of an odd statement I know but I think so.

 

[sir-pinski]

Thank you for sharing this formulation of Cantor's diagonal method! It's always interesting to see different ways of presenting mathematical concepts. Your explanation is clear and easy to follow, and I appreciate the effort you put into breaking down the steps of the proof. It's amazing to think about how such a simple concept can have such far-reaching implications in mathematics. Cantor's diagonal method is definitely one of the most elegant and powerful arguments in set theory.