Y×X

Y×X, also known as the Cartesian product of sets Y and X, is a fundamental concept in set theory and mathematics. It is denoted as Y×X and represents the set of all ordered pairs (y, x) where y is an element of Y and x is an element of X. The Cartesian product is named after the French mathematician René Descartes, who introduced the concept in the context of analytic geometry.

The Cartesian product Y×X is defined as follows: for any sets Y and X, Y×X = {(y, x) | y ∈ Y and x ∈ X}. This means that each element in the Cartesian product is an ordered pair, where the first element comes from set Y and the second element comes from set X. The order of the elements in the pair is significant, as (y, x) is generally not equal to (x, y) unless y = x and Y = X.

The Cartesian product has several important properties. It is associative, meaning that (Y×X)×Z is equivalent to Y×(X×Z) for any sets Y, X, and Z. It is also distributive over the union of sets, meaning that Y×(X ∪ Z) is equivalent to (Y×X) ∪ (Y×Z) for any sets Y, X, and Z.

In the context of geometry, the Cartesian product of two real number sets, denoted as ℝ×ℝ, represents the Cartesian plane, which is a two-dimensional coordinate system. Each point in the plane can be represented as an ordered pair (x, y) of real numbers.

The Cartesian product is a versatile concept that finds applications in various areas of mathematics, including topology, algebra, and geometry. It provides a way to combine elements from different sets and study their relationships.