Question about power sets and cartesian product

[dustbin]

Let A={1, 2} and B={∅}. First, I find the power set of A and the power set of B:

P(A)= { ∅, {1}, {2}, {1, 2} }
P(B)= { ∅, {∅} }

I believe the power sets are correct. I'm still new to the concept of power sets. Anyway, my main question is regarding cartesian product of power sets. I'm asked to find P(A)xP(B). I'm a bit confused when doing this operation due to the null set and set containing a null set.

This is my attempt:

P(A)xP(B)= { ∅, ({1}, {∅}), ({2}, {∅}), ({1,2}, {∅}) }

From my understanding, any nonempty set A multiplied by ∅ is Ax∅=∅. Is my answer correct?

 

[STEMucator]

Yes your answer is correct.

 

[tiny-tim]

hi dustbin!

From my understanding, any nonempty set A multiplied by ∅ is Ax∅=∅. Is my answer correct?

nooo … a product space is not multiplication

every element of a product space is an (ordered) set of two elements, one from each space

either or both of those elements can be ∅

(generally, the number of elements in X x Y is the number in X times the number in Y)

 

[dustbin]

Thank you for the responses!
Would you mind elaborating a bit more on ∅, tiny-tim? After some more reading, I can see why I am wrong for calling this multiplication. However, I am not sure about cartesian products involving ∅. For instance...

With P(A)xP(B), the "first" ordered pairs would be (∅, ∅), (∅, {∅}), ({1}, ∅). All of these are simply ∅. I can understand why the first would just be ∅, but I'm a little confused about the last one.

Pardon my ignorance on the matter. This is my first foray into the subject of sets.

 

[tiny-tim]

With P(A)xP(B), the "first" ordered pairs would be (∅, ∅), (∅, {∅}), ({1}, ∅). All of these are simply ∅.

no

you're confusing ∅ x B with {∅} x B

∅ has no elements, {∅} has one element​

∅ x B is {(x,y) : x ε ∅ and y ε B} … obviously, there's no such x, so there's no such (x,y), ie ∅ x B = ∅

{∅} x B is {(x,y) : x ε {∅} and y ε B} … obviously, there's exactly one such x, it's ∅, so {∅} x B = {(∅,y) : y ε B} … it has the same number of elements as B

if B contains only one element, say b, then ∅ x B = ∅ x {b} = (∅,b)

in particular, if b = ∅ (so B = {∅}), then ∅ x B = ∅ x ∅ = (∅,∅)​

it doesn't matter what the elements of a set are called …

if A has four elements, we can call them ∅,b,c,d or 1,2,3,4 or Lucy,Ricky,Fred,Ethel …

∅ is just as much a member of the set as Lucy is!

 

[dustbin]

Thank you tinytim. That's exactly what I was looking for. That helped immensely!