Cartesian Product

Cartesian Product

The Cartesian Product of two sets ( A ) and ( B ), denoted as ( A \times B ), is the set of all ordered pairs where the first element comes from ( A ) and the second comes from ( B ).

---

Definition

The Cartesian product of two sets ( A ) and ( B ) is defined as:

$$A\times B=\{(a,b)|a\in A,b\in B\}$$

This means that each element in set ( A ) is paired with every element in set ( B ).

---

Example

Consider two sets ( A = {1, 2} ) and ( B = {x, y} ).

$$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$$

Key Insight

The Cartesian product is not commutative: $A\times B\ne B\times A$. This means that:

$A\times B\ne B\times A$

In particular:

$A\times B\ne B\times A\text{ unless }A=B$

---

Generalization

The concept of Cartesian product can be extended to more than two sets. For example, the Cartesian product of three sets ( A ), ( B ), and ( C ) is:

$$A\times B\times C=\{(a,b,c)|a\in A,b\in B,c\in C\}$$

If ( A = {1, 2}, B = {x}, C = {y, z} ), the product is:

$$A\times B\times C=\{(1,x,y),(1,x,z),(2,x,y),(2,x,z)\}$$

Visualizing the Cartesian Product

• For ( A \times B ), imagine a grid where each row corresponds to an element of ( A ) and each column to an element of ( B ). The intersections represent the pairs.
• For higher dimensions, the visualization extends to cubes, and hypercubes for further sets.