Help With Find The Cardinality of a Power Set of a Cartesian Product

[daneault23]

Homework Statement

Suppose that A and B are finite sets.
What is |P(AxB)|? Meaning what is the cardinality of the power set of a cartesian product of the sets A and B.

Homework Equations

|AxB|=|A| * |B| since A and B are finite sets
Power set of a set is the set of all subsets of that set, including the empty set and the set itself
There are 2^|A| subsets for a set A when A is finite

The Attempt at a Solution

Since A and B are finite sets, we have |AxB|=|A| * |B|. Now the power set of (AxB) is the set of all its subsets, including the empty set and the set AxB itself. Since A and B are both finite sets, there is also a finite number of subsets of (AxB). By letting C=AxB, there are exactly 2^|C| subsets. Thus |P(AxB)|=2^|AxB|=2^(|A| * |B|)

This is what I have.

 

[jbunniii]

Looks fine to me.

 

[Dick]

That's just fine.

 

[daneault23]

I have a similar question asking what the |P(A)xP(B)| is. Can I ask that in this thread, or do I have to start another thread?

 

[Dick]

I have a similar question asking what the |P(A)xP(B)| is. Can I ask that in this thread, or do I have to start another thread?

You never HAVE to start another thread. It's usually recommended if the problem is unrelated but this one isn't. Ask here.

 

[daneault23]

The question asks what is |P(A)xP(B)|. This is asking what is the cardinality of the cartesian product of the power set of A and power set of B. I'm having some trouble deciphering this.

I let C=P(A) and D=P(B). Then, |CxD|=|C| * |D| since once again A and B are both finite sets, meaning their respective power sets, C and D, are also finite sets.

So, |P(A)xP(B)|=2^C * 2^D

That is what I have.

 

[Dick]

No problem again. That's correct.

 

[daneault23]

No problem again. That's correct.

Dick, are you saying that the syntax or form looks correct, or do you believe that is the correct answer?

So simplyifying it, it would be |P(A)xP(B)|=2^|C| * 2^|D|=2^2^|A| * 2^2^|B|

 

[Dick]

Dick, are you saying that the syntax or form looks correct, or do you believe that is the correct answer?

So simplyifying it, it would be |P(A)xP(B)|=2^|C| * 2^|D|=2^2^|A| * 2^2^|B|

No, I was reading too fast. It's 2^|A|*2^|B|. |C|=2^|A| and |D|=2^|B|. |CxD|=|C||D|.