Proof that the cardinality of a power set depends only on the cardinality of the underlying set.

Question

In all the textbooks I have seen, there has been no proof that the cardinality of the power set $P(A)$ of the set $A$ depends only on the cardinality of $A$, not on the set $A$ itself. What is the proof that it depends only on the cardinality of $A$? It is a detail that needs to be proven, but no book I have seen proves it.

but no book I have seen proves it --- Doesn't it follow from the fact that cardinal exponentiation is well defined (proved in many books) and the fact that $2^{\text{card }A} = \text{card }P(A)$ (proved in nearly every book, and also here)? — Dave L. Renfro, Jun 06 '22 at 20:52

score 3 · Accepted Answer · answered Jun 06 '22 at 20:51

(Well, if the question is going to exist there should be an Answer...)

The proof is not written out because it is so trivial.

Suppose $f:A\to B$ is a bijection. Define $F:P(A)\to P(B)$ by $$F(E)=\{f(x):x\in E\}.$$It's easy to show that $F$ is a bijection.

Proof that the cardinality of a power set depends only on the cardinality of the underlying set.

1 Answers1