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Carrie233

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## Homework Statement

Prove:

If A and B each have at least two elements, then not every element of P(A×B) has the form A1 ×B1 for some A1 ∈ P(A)and B1 ∈ P(B).

## Homework Equations

## The Attempt at a Solution

Suppose A = {1, 2}, B = {3, 4}.

AXB = {(1,3), (1,4), (2,3), (2,4)}

P(A) = {{1}, {2}, {1,2}, ø}；P(B) = {{3}, {4}, {3,4}, ø}

Since A1 ∈ P(A), A1 could be {1}, {2}, {1,2}, or ø; similarly, B1 could be {3}, {4}, {3,4}, or ø

P(AXB) = {{(1,3)}, {(1,4)}, {(2,3)}, {(2,4)}, {(1,3),(1,4)}, ... {(1,3), (1,4), (2,3), (2,4)} }

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