The Cartesian product A × B is defined as the set of all possible ordered pairs (a, b) where a is an element of set A and b is an element of set B. For example, with A = {1, 2, 3} and B = {cow, sheep}, the correct representation of A × B is {(1, cow), (1, sheep), (2, cow), (2, sheep), (3, cow), (3, sheep)}. It is important to note that the order of elements in the pairs matters, making Cartesian products non-commutative; (1, cow) is not the same as (cow, 1). The discussion clarified the distinction between "all possible ordered pairs" and "all ordered pairs," emphasizing that the Cartesian product includes distinct pairs based on the order of elements. Understanding these concepts helps in grasping the fundamentals of set theory and Cartesian products.
