Cartesian Product of Intersections/Corollary 2

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Corollary to Cartesian Product of Intersections

$\paren {A \times B} \cap \paren {B \times A} = \paren {A \cap B} \times \paren {A \cap B}$


Proof

Take the result Cartesian Product of Intersections:

$\paren {S_1 \cap S_2} \times \paren {T_1 \cap T_2} = \paren {S_1 \times T_1} \cap \paren {S_2 \times T_2}$

Put $S_1 = A, S_2 = B, T_1 = B, T_2 = A$:

\(\displaystyle \) \(\) \(\displaystyle \paren {A \times B} \cap \paren {B \times A}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {A \cap B} \times \paren {B \cap A}\) Cartesian Product of Intersections
\(\displaystyle \) \(=\) \(\displaystyle \paren {A \cap B} \times \paren {A \cap B}\) Intersection is Commutative

$\blacksquare$


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