Proving Cardinality of P(S) > S

[Mr Davis 97]

Homework Statement

Prove that the cardinality of $P(S)$ is greater than the cardinality of S, where S is any set.

Homework Equations

The Attempt at a Solution

It would seem that we could simply define $T: S \rightarrow P(S)$ such that $T(s) = \{s \}$. This is clearly an injection, so $|S| \le |P(S)|$. That seemed too easy though, so I feel like I am doing something wrong.

 

[S.G. Janssens]

That seemed too easy though, so I feel like I am doing something wrong.

I think you are asked to prove strict inequality of cardinalities.

 

[Mr Davis 97]

I think you are asked to prove strict inequality of cardinalities.

So I have to show that there exists no bijection between $S$ and its powerset?