Cartesian Products

Set Theory · Axiom Academy

Creating ordered pairs from two sets Before studying Cartesian products, we need to understand ordered pairs . An ordered pair is a pair of objects where order matters. Two ordered pairs are equal if and only if both components match: Consider coordinates on a plane: is the point 3 units right, 5 units up is the point 5 units right, 3 units up In sets, because sets don't have order. But in ordered pairs, because order matters! The Cartesian product of sets A and B, written , is the set of all ordered pairs where the first element comes from A and the second from B: We get 6 pairs = 2 × 3 (size of A × size of B). We can visualize this as a grid: Each element of A can pair with each element of B, giving |A| × |B| total pairs. Unlike union and intersection, the Cartesian product is not commutative : These are different sets! The pairs have components in different order. The Cartesian product of a set with itself is denoted : This is the set of all ordered pairs of elements from A. Coordinate systems: (the 2D plane) Databases: Relations are subsets of Cartesian products Functions: A function from A to B is a subset of A × B Ordered pairs (a,b) have order: is all pairs with first from A, second from B Foundation for defining relations and functions

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