Is the Cartesian product of ( A × B ) c a subset of A...

Ethen Blackwell

Ethen Blackwell

Answered

2022-07-25

Is the Cartesian product of ( A × B ) c a subset of A c × B c ?

Answer & Explanation

nezivande0u

nezivande0u

Expert

2022-07-26Added 16 answers

Suppose A P and B Q. Then the Cartesian product A × B is a subset of the Cartesian product P × Q.
We know that ( x , y ) A × B if both x A and y B are true, which fails if either x ≠∈ A or y ≠∈ B or both. Thus, ( A × B ) c = { ( x , y ) P × Q : ( x , y ) ≠∈ A × B } = ( A c × Q ) ( P × B c )
However, the sets ( A c × Q ) and ( P × B c ) are not disjoint sets so that the pairs where both elements are not in the corresponding subsets are being double counted. This can be corrected as under:
( A × B ) c = ( A c × B ) ( A × B c ) ( A c × B c ). Thus ( A × B ) c is not a subset of A c × B c On the contrary, A c × B c is a subset of ( A × B ) c

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