The Cartesian product of two sets A and B, denoted as A × B, is a mathematical operation that combines every element of set A with every element of set B. In other words, it creates a new set that contains all possible ordered pairs (a, b), where a is an element of A and b is an element of B.
The Cartesian product contains the following knowledge points:
The formula for the Cartesian product of sets A and B is as follows: A × B = {(a, b) | a ∈ A, b ∈ B}
To apply the Cartesian product formula, follow these steps:
The symbol for the Cartesian product of sets A and B is ×. It is written as A × B.
There are several methods to find the Cartesian product of sets A and B:
Let set A = {1, 2} and set B = {a, b}. Find the Cartesian product A × B. Solution: A × B = {(1, a), (1, b), (2, a), (2, b)}
Consider set A = {x, y} and set B = {1, 2, 3}. Determine the Cartesian product A × B. Solution: A × B = {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)}
Given set A = {red, green} and set B = {circle, square}. Calculate the Cartesian product A × B. Solution: A × B = {(red, circle), (red, square), (green, circle), (green, square)}
Q: What is the purpose of finding the Cartesian product of sets A and B? A: The Cartesian product is used in various mathematical concepts, such as relations, functions, and combinatorics. It helps in understanding the relationships between elements of different sets and finding all possible combinations.
Q: Can the Cartesian product be applied to more than two sets? A: Yes, the Cartesian product can be extended to more than two sets. For example, the Cartesian product of sets A, B, and C would be denoted as A × B × C.
Q: Is the Cartesian product commutative? A: No, the Cartesian product is not commutative. In other words, A × B is not always equal to B × A. The order of sets matters in the Cartesian product.