Set theory MOC

Ordered pair

An ordered pair is a construction satisfying the fundamental property set

(𝑎1,𝑏1)=(𝑎2,𝑏2)⟺[𝑎1=𝑎2]∧[𝑏1=𝑏2]

the set of all ordered pairs from a given pair of sets forms the cartesian product. One may then define an ordered 𝑛-tuple by (𝑎,𝑏,𝑐) =((𝑎,𝑏),𝑐), &c. Compare this with the related universal property of the categorical product.

Construction

Within ZF the typical model, due to Kazimierz Kuratowski, is as follows

(𝑎,𝑏)=𝐾{{𝑎},{𝑎,𝑏}}

which satisfies the fundamental property.

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