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

P(A)|=2^|A| and |P(B)|=2^|B|. Thus, |P(A)xP(B)|=|P(A)|*|P(B)|=2^|A|*2^|B|. In summary, the cardinality of the power set of the cartesian product of two finite sets A and B is equal to 2^|A|*2^|B|. This is because the power set of a set is the set of all its subsets, including the empty set and the set itself. Since A and B are finite sets, their respective power sets are also finite sets. Therefore, the cardinality of the power set of (AxB) is equal to the
  • #1
daneault23
32
0

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.
 
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  • #3
That's just fine.
 
  • #4
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?
 
  • #5
daneault23 said:
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.
 
  • #6
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.
 
  • #7
No problem again. That's correct.
 
  • #8
Dick said:
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|
 
  • #9
daneault23 said:
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|.
 
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Related to Help With Find The Cardinality of a Power Set of a Cartesian Product

1. What is the cardinality of a power set?

The cardinality of a power set is the number of elements in the set of all subsets of a given set. It is denoted by |P(S)|, where S is the original set.

2. How do you find the cardinality of a power set?

To find the cardinality of a power set, you can use the formula 2^n, where n is the number of elements in the original set. Alternatively, you can list out all the possible subsets and count them.

3. What is a Cartesian product?

A Cartesian product is a mathematical operation that combines two sets to create a new set. The resulting set contains all possible ordered pairs of elements from the two original sets.

4. How do you find the Cartesian product of two sets?

To find the Cartesian product of two sets, you can list out all possible ordered pairs by combining each element of the first set with each element of the second set. For example, if set A = {1,2} and set B = {a,b}, the Cartesian product of A and B would be {(1,a), (1,b), (2,a), (2,b)}.

5. How do you find the cardinality of a power set of a Cartesian product?

To find the cardinality of a power set of a Cartesian product, you can use the formula 2^(m*n), where m and n are the cardinalities of the two original sets. Alternatively, you can list out all possible subsets of the Cartesian product and count them.

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