What is the Cartesian Product?
The Cartesian product of two sets, A and B, denoted A × B (read as "A cross B"), is a new set containing all possible ordered pairs (a, b) where the first element 'a' comes from set A and the second element 'b' comes from set B. It systematically pairs every element of the first set with every element of the second set.
Key Principles and Notation
If set A has 'm' distinct elements and set B has 'n' distinct elements, then their Cartesian product A × B will contain m × n ordered pairs. The crucial aspect is that these are *ordered* pairs, meaning (a, b) is generally different from (b, a) unless a equals b. This concept extends to the Cartesian product of three or more sets, forming ordered tuples (e.g., A × B × C results in (a, b, c) tuples).
A Practical Example
Consider two simple sets: A = {apple, banana} and B = {red, green}. The Cartesian product A × B would be {(apple, red), (apple, green), (banana, red), (banana, green)}. Here, set A has 2 elements and set B has 2 elements, so A × B has 2 × 2 = 4 ordered pairs.
Importance and Applications
The Cartesian product is a foundational concept in mathematics, used to rigorously define relations, functions, and the framework for coordinate geometry. For example, the familiar two-dimensional Cartesian coordinate plane (like the x-y plane) is formally defined as the Cartesian product of the set of all real numbers (ℝ) with itself, ℝ × ℝ, representing every possible point (x, y) in that plane.