Editing Cartesian Product

= Cartesian Product - Definition, Explanation, and Examples =

The '''Cartesian Product''' is an operation used in mathematics to combine two sets and form a new set made of ordered pairs. This concept is widely used in set theory, coordinate geometry, and computer science.

== Definition ==

If <math>A</math> and <math>B</math> are two sets, the '''Cartesian product''' of <math>A</math> and <math>B</math> is the set of all ordered pairs where:

- The first element is from set <math>A</math>
- The second element is from set <math>B</math>

It is denoted as:

<math>
A \times B = \{ (a, b) \mid a \in A,\, b \in B \}
</math>

== Important Points ==

* The order in each pair matters. That is, <math>(a, b) \ne (b, a)</math> unless <math>a = b</math>.
* If either set is empty, the Cartesian product is also empty:  
  <math>A \times \emptyset = \emptyset</math> and <math>\emptyset \times B = \emptyset</math>
* The total number of ordered pairs in <math>A \times B</math> is:  
  <math>n(A \times B) = n(A) \times n(B)</math>

== Step-by-Step Example 1 ==

Let:  
<math>A = \{1, 2\}</math>  
<math>B = \{x, y\}</math>

Step 1: Take each element of set <math>A</math>  
Step 2: Pair it with each element of set <math>B</math>

<math>
A \times B = \{(1, x), (1, y), (2, x), (2, y)\}
</math>

There are 4 ordered pairs because <math>2 \times 2 = 4</math>.

== Step-by-Step Example 2 ==

Let:  
<math>C = \{a, b\}</math>  
<math>D = \{1, 2, 3\}</math>

Then:  
<math>
C \times D = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}
</math>

<math>n(C \times D) = 2 \times 3 = 6</math> ordered pairs.

== Example 3: Cartesian Product with Itself ==

Let:  
<math>E = \{0, 1\}</math>

Then:  
<math>
E \times E = \{(0, 0), (0, 1), (1, 0), (1, 1)\}
</math>

This is also known as a set of points in a 2D space — for example, like grid points in coordinate geometry.

== Visual Representation ==

If you consider <math>A = \{1, 2\}</math> and <math>B = \{3, 4\}</math>, then <math>A \times B</math> can be visualized as points in a table or plane:

|| **B = 3** || **B = 4** ||
|------------|-------------|
| (1, 3)     | (1, 4)      |
| (2, 3)     | (2, 4)      |

== Applications ==

* Coordinate geometry (e.g., the Cartesian plane)  
* Relations and functions  
* Computer science and databases  
* Logic and discrete mathematics

== Summary ==

The Cartesian product of two sets combines all elements from both sets into ordered pairs. It forms the basis of many mathematical concepts like relations, functions, and coordinates.

[[Category:Set Theory]]
[[Category:Ordered Pairs]]
[[Category:Mathematics]]
[[Category:Cartesian Product]]