Example Showing (A x B) U (X x Y) ≠ (A U X) x (B U Y)

[laminatedevildoll]

I have to find an example illustrating the the equation
(A x B) U (X x Y) = (A U X) x (B U Y) is not valid for all choices of sets A,B,X,Y.

I started out by saying that

let (x,y) be in A and B or
let (x,y) be in X and Y

Since x is in A and y is in B
--> (x,y) = (A x B)
Since x is in X and y is in Y
--> (x,y) = (X x Y)

Can I go on to say that since x is in A and x is in A
x is a subset of (A U X)?

How do I prove that this is not true?

 

[Muzza]

You're thinking much too hard. Pick A, B, X, Y as singletons, etc.

 

[BicycleTree]

(A x B) U (X x Y) = (A U X) x (B U Y)

(A U X) x (B U Y) can never be of prime cardinality if A U X and B U Y are not singletons.

When can (A x B) U (X x Y) be of prime cardinality?

 

[laminatedevildoll]

Is it when it's an intersection? Okay, so I will have to find a counterexample for it. Also, I was wondering if it's wise to start off by trying to prove the example, and then find out that its wrong, and then to find a counterexample for it.