Union of Union of Cartesian Product with Empty Factor

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Theorem

Let $A$ and $B$ be sets such that either $A = \O$ or $B = \O$.

Let the ordered pair $\tuple {a, b}$ be defined using the Kuratowski formalization:

$\tuple {a, b} := \set {\set a, \set {a, b} }$


Then:

$\ds \bigcup \bigcup \paren {A \times B} = A \cup B \iff A = B = \O$

where:

$\cup$ denotes union
$\times$ denotes Cartesian product.


That is, if either $A$ or $B$ is empty:

$\ds \bigcup \bigcup \paren {A \times B} = A \cup B$

holds if and only if they are both empty


Proof

Let $A = \O$ or $B = \O$.

From Cartesian Product is Empty iff Factor is Empty:

$A \times B = \O$

Hence from Union of Empty Set:

$\ds \bigcup \bigcup \paren {A \times B} = \O$

However, from Union is Empty iff Sets are Empty:

$A \cup B = \O \iff A = \O \text { and } B = \O$

The result follows.

$\blacksquare$


Also see


Sources