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The Cartesian product of a set and the empty set

adnauseum profile image Samuel Kendrick ・1 min read

The cartesian product is the product of two sets. The resulting product is a set of pairs, one element belonging to AA and one element belonging to BB .

This can expressed as:

A×B={(x,y):xA,yB} {A \times B = \{ (x,y) : x \in A, y \in B \} }

So what happens if we try to take the cartesian product of a set and the empty set? A×{A \times \emptyset} ?

We end up with: A×={A \times \emptyset} = \emptyset . This is because we try to build a set like this: A×B={(x,y):xA,yB}{A \times B = \{ (x,y) : x \in A, y \in B \} } , but BB is empty.

It is impossible to create this resulting set of pairs because one of the sets is the empty set. As such, not a single element can be produced from BB , so our resulting set is the empty set.

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