What is the definition of a Cartesian product of two sets?

• A. The set of all elements common in both sets
• B. The set of all possible ordered pairs from two sets
• C. Two sets combined into a single set with no repetitions
• D. The operation of finding the intersection of two sets

Answer: (B)

The Cartesian product of two sets is defined as the set of all possible ordered pairs that can be formed where the first element of each pair comes from the first set, and the second element comes from the second set. This operation is denoted as A × B, where A and B are the two sets.

For example, if you have set A = {1, 2} and set B = {x, y}, the Cartesian product A × B would yield the set { (1, x), (1, y), (2, x), (2, y) }. Each element of set A pairs with every element of set B, resulting in all possible combinations of these elements as ordered pairs.

This concept is fundamental in set theory and is used frequently in various branches of mathematics, computer science, and related fields, particularly in situations involving relational data or multi-dimensional data structures. Understanding the Cartesian product is essential for further exploration of functions, relations, and more complex set operations.