Ordered pair

An ordered pair is a pair of objects with a designated first position and second position. Reversing the positions generally produces a different pair.

Set: A set is a collection of distinct objects treated as a single mathematical object. Ordered-pair constructions use sets as their basic ingredients.
Kuratowski ordered pair: The Kuratowski ordered pair of x and y is the set {{x}, {x, y}}. This standard set-theoretic encoding makes the first and second positions recoverable.
Coordinate geometry: Coordinate geometry studies geometric objects using numerical coordinates. A point in the plane is represented by an ordered pair of coordinates.
Unordered pair: An unordered pair is a collection of two objects in which their order does not distinguish the pair. Unlike an ordered pair, swapping its entries leaves the same collection.
Cartesian product: The Cartesian product of two sets is the set of all ordered pairs whose first and second entries come from the respective sets. It organizes ordered pairs into a set according to the allowed entries.
Wiener ordered pair: The Wiener ordered pair of x and y is the set {{{x}, ∅}, {{y}}}. It is an early set-theoretic encoding of positional order.
Function: A function assigns exactly one output to each element of its domain. Its graph is a set of ordered pairs linking inputs to outputs.
Set union: The union of two sets contains every element belonging to either set. Union discards positional roles, unlike an ordered pair.
Equality: Equality is the relation stating that two expressions denote the same object. Its pair-specific rule requires both corresponding positions to match.
Axiom of extensionality: The axiom of extensionality states that sets with the same members are equal. It helps establish that the Kuratowski encoding obeys the intended equality rule.

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