Understanding the Cartesian Product of Sets A and B: A × B vs B × A

When dealing with set operations, the Cartesian product is a fundamental concept. This article explores the differences between the Cartesian products A × B and B × A when A {1, 2, 3} and B {a, b}.

Introduction to Cartesian Product

Let's start by defining the Cartesian product. For two sets A and B, the Cartesian product A × B is defined as the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. Similarly, B × A is the set of all ordered pairs (b, a) where b is an element of B and a is an element of A.

Example Sets and Their Cartesian Products

Given the sets A {1, 2, 3} and B {a, b}, we can calculate A × B and B × A as follows:

A × B

A × B is the set of all ordered pairs (a, b) where a is from A and b is from B:

A × B {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

B × A

B × A is the set of all ordered pairs (b, a) where b is from B and a is from A:

B × A {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

Initially, these two sets may look similar, but let's explore their elements more closely.

Differences Between A × B and B × A

Upon closer inspection, we can see that the elements in A × B and B × A are not the same. For instance, in A × B, the elements are ordered pairs where the first element is from A and the second is from B. In B × A, the elements are ordered pairs where the first element is from B and the second is from A.

Example Based on Given Sets

Let's consider the specific sets you provided:

A {1, 2, 3}

B {a, b}

Then, the Cartesian products are:

A × B {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

B × A {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

As you can see, the elements in A × B and B × A are not the same. For example, (1, a) is in A × B but not in B × A, and (a, 1) is in B × A but not in A × B.

Equal Cartesian Products

The Cartesian products A × B and B × A are only equal when the sets A and B are the same. In other words, A × B B × A if and only if A B.

Conclusion

In summary, the Cartesian products of sets depend crucially on the order of the elements. For the sets A {1, 2, 3} and B {a, b}, A × B and B × A are not the same. Moreover, the Cartesian products are equal only if the sets themselves are identical.

If you have any further questions or need more detailed information, please feel free to ask!