# Help with Cartesian product of power sets please

#### chuciulla93

I hope I'm not breaking any rules by posting this, but just came across a homework problem that I was unsure on.
What is the Cartesian product of A and the power set of B if A={1} and B={}?
I know that the Cartesian product of A x B under these cases is the empty set, but it's not the same thing.
So far I've tried looking at it a few different ways: {1} x {{}}= ? being one
Thank you

PS: I'm new here so please be gentle (Angel)

#### Plato

MHF Helper
So far I've tried looking at it a few different ways: {1} x {{}}= ? being one

$$\displaystyle \{1\}\times\{\emptyset\}$$ has one element. What is the ordered pair?

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• chuciulla93 and (deleted member)

#### chuciulla93

Would it be: { (1,{}), ({},1) } ?
And yes the pair is $$\displaystyle (1,\emptyset}).$$

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Thank you

#### Plato

MHF Helper
And yes the pair is $$\displaystyle (1,\emptyset}).$$
The power set $$\displaystyle \mathcal{P}(\emptyset)=\{\emptyset\}$$
The power set $$\displaystyle \mathcal{P}(\{1\})=\{\emptyset,~\{1\}\}$$
Therefore $$\displaystyle \mathcal{P}(\{1\})\times \mathcal{P}(\emptyset)=\{(\emptyset,\emptyset),~ ( \{ 1\},\emptyset)\}$$: that is two elements.